GCSE Maths: Trigonometric graphs (Higher)
Part of the GCSE Maths functions and advanced graphs chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Trigonometric graphs
In Chapter 20 you used sin, cos and tan for angles in right-angled triangles, so the angles were between 0° and 90°. A calculator will also give sin, cos and tan of any angle, even 210° or −40°. When we plot these values against the angle we get three famous graphs. Always make sure your calculator is in degree mode.
The sine graph, y = sin x. It starts at 0, rises to a maximum of 1 at 90°, falls back to 0 at 180°, reaches a minimum of −1 at 270° and returns to 0 at 360°. After that the pattern repeats. The length of one complete pattern is the period, which is 360° for sine. All the values lie between −1 and 1.
The cosine graph, y = cos x. It has the same wave shape but starts at its maximum: cos 0° = 1, then 0 at 90°, a minimum of −1 at 180°, 0 at 270° and back to 1 at 360°. The period is again 360°. The cosine graph is the sine graph moved 90° to the left: cos x = sin(x + 90°). This is exactly the transformation f(x + 90) from the last section.
The tangent graph, y = tan x. This one looks very different. tan x = sin x ÷ cos x, so it is not defined where cos x = 0, that is at 90° and 270°. There the graph has vertical asymptotes (lines it gets closer and closer to but never touches). Between asymptotes the graph rises from very negative to very positive. It is 0 at 0°, 180° and 360°, and it equals 1 at 45°. Its period is 180°, and it has no maximum or minimum value.
It helps to know the key values. They are given to 2 decimal places here; some are exact (0, ±1, ±0.5).
| x | 0° | 30° | 60° | 90° | 120° | 150° | 180° | 210° | 240° | 270° | 300° | 330° | 360° |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| sin x | 0.00 | 0.50 | 0.87 | 1.00 | 0.87 | 0.50 | 0.00 | −0.50 | −0.87 | −1.00 | −0.87 | −0.50 | 0.00 |
| cos x | 1.00 | 0.87 | 0.50 | 0.00 | −0.50 | −0.87 | −1.00 | −0.87 | −0.50 | 0.00 | 0.50 | 0.87 | 1.00 |
| tan x | 0.00 | 0.58 | 1.73 | – | −1.73 | −0.58 | 0.00 | 0.58 | 1.73 | – | −1.73 | −0.58 | 0.00 |
Using symmetry. Look at the sine graph: the hump between 0° and 180° is symmetrical about 90°, so sin 150° = sin 30° = 0.5. The part between 180° and 360° is the same hump turned upside down, so sin 210° = −sin 30° = −0.5. For any angle, use the graph to decide whether the value is positive or negative, and use the related acute angle for the size.
Solving equations such as sin x = 0.3. The solutions are where the horizontal line y = 0.3 crosses the graph. The calculator button gives only the first solution, 17.5°. Use the symmetry of the graph for the others: for sin, the second solution is 180° − the first; for cos it is 360° − the first; for tan, add 180°. Then add or subtract 360° (180° for tan) to find solutions in the required range.
Transformations. The ideas of the last section work on these graphs too. y = 2 sin x has the same period but amplitude 2 (maximum 2, minimum −2). y = sin 2x completes a wave in 180° instead of 360°, so its period is 360° ÷ 2 = 180°. y = sin x + 1 is moved up 1, and y = −sin x is turned upside down.
Key ideas
- sin x and cos x: period 360°, maximum 1, minimum −1. cos x starts at 1; sin x starts at 0.
- tan x: period 180°, asymptotes at 90° and 270°, passes through (45°, 1), no maximum or minimum.
- Use symmetry: sin 150° = sin 30°, cos 120° = −cos 60°, tan 135° = −tan 45°. Decide the sign from the graph.
- To solve sin x = k: first solution from the calculator, second solution 180° − first (cos: 360° − first; tan: first + 180°), then add or subtract full periods to reach the range.
- y = a sin x: amplitude a. y = sin bx: period 360°/b. y = sin x + d: shift up d.
Method. Use the symmetry of the sin graph: the size of the value is the same as for the related acute angle, and the sign comes from the quarter.
- Find how 210° is related to 30°. Use the cycle of 360°: angles in the first quarter are 0° to 90°, second 90° to 180°, third 180° to 270°, fourth 270° to 360°.210° = 180° + 30° (a third-quarter angle)
- Decide whether the value is positive or negative. Look at the graph of sin x: in the third quarter the curve is below the x-axis.sin 210° is negative.
- Use the same size with the correct sign. The graph is a mirror image (or turned image) of the first quarter, so the size of the value is the same.sin 210° = −0.5 = −0.5
Check. On a calculator, sin 210° = −0.500, which matches −0.5. ✓
Watch out. Do not forget the sign. Check whether the curve is above or below the axis at that angle.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the sin graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of sin x = 0.3 are the x-values where the horizontal line y = 0.3 crosses the graph of y = sin x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.sin−1(0.3) = 17.5°
- Use the symmetry of the graph to find the other solutions. The sine graph is symmetrical about x = 90° (and x = 270°). So the second solution in the same cycle is 180° − the first.180 − 17.5 = 162.5
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.17.5, 162.5
Check. Put each answer into the calculator: sin 17.5° = 0.30; sin 162.5° = 0.30 (rounding to 2 d.p. should give about 0.30). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the cos graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of cos x = −0.4 are the x-values where the horizontal line y = −0.4 crosses the graph of y = cos x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.cos−1(−0.4) = 113.6°
- Use the symmetry of the graph to find the other solutions. The cosine graph is symmetrical about x = 0° and x = 180°. So the second solution is the negative of the first, then add 360° to bring it into the range.−113.6 + 360 = 246.4
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 360°. Keep only answers from 0° to 360°.113.6, 246.4
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.113.6, 246.4
Check. Put each answer into the calculator: cos 113.6° = −0.40; cos 246.4° = −0.40 (rounding to 2 d.p. should give about −0.40). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the tan graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of tan x = 2 are the x-values where the horizontal line y = 2 crosses the graph of y = tan x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.tan−1(2) = 63.4°
- Use the symmetry of the graph to find the other solutions. The tangent graph repeats every 180°, so the other solutions are found by adding (or subtracting) 180°.63.4 + 180 = 243.4
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 180°. Keep only answers from 0° to 360°.63.4, 243.4
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.63.4, 243.4
Check. Put each answer into the calculator: tan 63.4° = 2.00; tan 243.4° = 2.00 (rounding to 2 d.p. should give about 2.00). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Method. Read the amplitude, the period and the middle line from the graph, then match them to y = a sin(bx) + d (or cos).
- Read the maximum and minimum values. The graph oscillates between a highest and lowest value.maximum = 2, minimum = −2
- Find the amplitude and the middle line. The amplitude is half the distance between the maximum and the minimum. The middle line is halfway between them.amplitude = (2 − (−2)) ÷ 2 = 2
middle line: y = (2 + (−2)) ÷ 2 = 0 - Read the period (the length of one full cycle). The standard sin graph has period 360°. If one cycle takes 120° then the angle is multiplied by 3.period = 120° ⇒ b = 360 ÷ 120 = 3
- Write the equation. Use y = a sin (bx) + d with a = 2, b = 3, d = 0.y = 2 sin 3x
Check. Test one point: when x = 0, the equation gives 0.00, and the graph passes through that value. ✓
Watch out. The period is the length of one whole cycle, not half of it.
Watch out
The calculator gives only one answer to sin x = 0.3, but there are two in each 360°. Use the graph (or symmetry) to find the others. Also check that the calculator is in degree mode, and for equations like 2 sin x = 1, divide by the 2 first to get sin x = 0.5.Exercise 25E Trigonometric graphs
Calculator allowed| x | 0° | 60° | 90° | 120° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|
| cos x |
Method. Type cos of each angle into the calculator, with the calculator set to degrees, and round to 1 decimal place.
- Check the calculator mode. It must show D or DEG. In radian mode all the answers would be wrong.
- Work out each value. Press cos, then the angle, then equals.cos 0° = 1.000 ≈ 1.0
cos 60° = 0.500 ≈ 0.5
cos 90° = 0.000 ≈ 0.0
cos 120° = −0.500 ≈ −0.5
cos 180° = −1.000 ≈ −1.0
cos 270° = 0.000 ≈ 0.0
cos 360° = 1.000 ≈ 1.0 - Write them in the table.
x 0° 60° 90° 120° 180° 270° 360° cos x 1.0 0.5 0.0 −0.5 −1.0 0.0 1.0
Check. The values match the shape of the cosine graph: 1, then down to −1 at 180°, then back up to 1. ✓
Watch out. cos 90° and cos 270° are exactly 0. A calculator may show a tiny number like 6 × 10−17; that is a rounding error.
Method. Read the standard graph.
- Maximum and minimum. The wave rises to 1 and falls to −1.maximum 1, minimum −1
- Period. One complete wave takes 360°.period 360°
Check. sin 90° = 1 and sin 270° = −1. ✓
Watch out. The period is the length of one complete cycle.
Method. Read the standard graph.
- Maximum and minimum. cos 0° = 1 and cos 180° = −1.maximum 1, minimum −1
- Period. One complete wave takes 360°.period 360°
Check. cos 0° = 1 and cos 180° = −1. ✓
Watch out. cos starts at its maximum.
Method. Read the standard graph.
- Maximum and minimum. The graph goes up to very large positive values and down to very large negative values at the asymptotes.no maximum or minimum
- Period. The pattern repeats every 180°.period 180°
Check. tan 45° = 1 and tan 225° = 1, which are 180° apart. ✓
Watch out. tan has asymptotes, so it has no highest or lowest value.
Method. Use the symmetry of the sin graph: the size of the value is the same as for the related acute angle, and the sign comes from the quarter.
- Find how 150° is related to 30°. Use the cycle of 360°: angles in the first quarter are 0° to 90°, second 90° to 180°, third 180° to 270°, fourth 270° to 360°.150° = 180° − 30° (a second-quarter angle)
- Decide whether the value is positive or negative. Look at the graph of sin x: in the second quarter the curve is above the x-axis.sin 150° is positive.
- Use the same size with the correct sign. The graph is a mirror image (or turned image) of the first quarter, so the size of the value is the same.sin 150° = 0.5 = 0.5
Check. On a calculator, sin 150° = 0.500, which matches 0.5. ✓
Watch out. Do not forget the sign. Check whether the curve is above or below the axis at that angle.
Method. Use the symmetry of the sin graph: the size of the value is the same as for the related acute angle, and the sign comes from the quarter.
- Find how 210° is related to 30°. Use the cycle of 360°: angles in the first quarter are 0° to 90°, second 90° to 180°, third 180° to 270°, fourth 270° to 360°.210° = 180° + 30° (a third-quarter angle)
- Decide whether the value is positive or negative. Look at the graph of sin x: in the third quarter the curve is below the x-axis.sin 210° is negative.
- Use the same size with the correct sign. The graph is a mirror image (or turned image) of the first quarter, so the size of the value is the same.sin 210° = −0.5 = −0.5
Check. On a calculator, sin 210° = −0.500, which matches −0.5. ✓
Watch out. Do not forget the sign. Check whether the curve is above or below the axis at that angle.
Method. Use the symmetry of the sin graph: the size of the value is the same as for the related acute angle, and the sign comes from the quarter.
- Find how 330° is related to 30°. Use the cycle of 360°: angles in the first quarter are 0° to 90°, second 90° to 180°, third 180° to 270°, fourth 270° to 360°.330° = 360° − 30° (a fourth-quarter angle)
- Decide whether the value is positive or negative. Look at the graph of sin x: in the fourth quarter the curve is below the x-axis.sin 330° is negative.
- Use the same size with the correct sign. The graph is a mirror image (or turned image) of the first quarter, so the size of the value is the same.sin 330° = −0.5 = −0.5
Check. On a calculator, sin 330° = −0.500, which matches −0.5. ✓
Watch out. Do not forget the sign. Check whether the curve is above or below the axis at that angle.
Method. Use the symmetry of the cos graph: the size of the value is the same as for the related acute angle, and the sign comes from the quarter.
- Find how 120° is related to 60°. Use the cycle of 360°: angles in the first quarter are 0° to 90°, second 90° to 180°, third 180° to 270°, fourth 270° to 360°.120° = 180° − 60° (a second-quarter angle)
- Decide whether the value is positive or negative. Look at the graph of cos x: in the second quarter the curve is below the x-axis.cos 120° is negative.
- Use the same size with the correct sign. The graph is a mirror image (or turned image) of the first quarter, so the size of the value is the same.cos 120° = −0.5 = −0.5
Check. On a calculator, cos 120° = −0.500, which matches −0.5. ✓
Watch out. Do not forget the sign. Check whether the curve is above or below the axis at that angle.
Method. Use the symmetry of the cos graph: the size of the value is the same as for the related acute angle, and the sign comes from the quarter.
- Find how 240° is related to 60°. Use the cycle of 360°: angles in the first quarter are 0° to 90°, second 90° to 180°, third 180° to 270°, fourth 270° to 360°.240° = 180° + 60° (a third-quarter angle)
- Decide whether the value is positive or negative. Look at the graph of cos x: in the third quarter the curve is below the x-axis.cos 240° is negative.
- Use the same size with the correct sign. The graph is a mirror image (or turned image) of the first quarter, so the size of the value is the same.cos 240° = −0.5 = −0.5
Check. On a calculator, cos 240° = −0.500, which matches −0.5. ✓
Watch out. Do not forget the sign. Check whether the curve is above or below the axis at that angle.
Method. Use the symmetry of the cos graph: the size of the value is the same as for the related acute angle, and the sign comes from the quarter.
- Find how 300° is related to 60°. Use the cycle of 360°: angles in the first quarter are 0° to 90°, second 90° to 180°, third 180° to 270°, fourth 270° to 360°.300° = 360° − 60° (a fourth-quarter angle)
- Decide whether the value is positive or negative. Look at the graph of cos x: in the fourth quarter the curve is above the x-axis.cos 300° is positive.
- Use the same size with the correct sign. The graph is a mirror image (or turned image) of the first quarter, so the size of the value is the same.cos 300° = 0.5 = 0.5
Check. On a calculator, cos 300° = 0.500, which matches 0.5. ✓
Watch out. Do not forget the sign. Check whether the curve is above or below the axis at that angle.
Method. Use the symmetry of the tan graph: the size of the value is the same as for the related acute angle, and the sign comes from the quarter.
- Find how 135° is related to 45°. Use the cycle of 360°: angles in the first quarter are 0° to 90°, second 90° to 180°, third 180° to 270°, fourth 270° to 360°.135° = 180° − 45° (a second-quarter angle)
- Decide whether the value is positive or negative. Look at the graph of tan x: in the second quarter the curve is below the x-axis.tan 135° is negative.
- Use the same size with the correct sign. The graph is a mirror image (or turned image) of the first quarter, so the size of the value is the same.tan 135° = −1 = −1
Check. On a calculator, tan 135° = −1.000, which matches −1. ✓
Watch out. Do not forget the sign. Check whether the curve is above or below the axis at that angle.
Method. Use the symmetry of the tan graph: the size of the value is the same as for the related acute angle, and the sign comes from the quarter.
- Find how 225° is related to 45°. Use the cycle of 360°: angles in the first quarter are 0° to 90°, second 90° to 180°, third 180° to 270°, fourth 270° to 360°.225° = 180° + 45° (a third-quarter angle)
- Decide whether the value is positive or negative. Look at the graph of tan x: in the third quarter the curve is above the x-axis.tan 225° is positive.
- Use the same size with the correct sign. The graph is a mirror image (or turned image) of the first quarter, so the size of the value is the same.tan 225° = 1 = 1
Check. On a calculator, tan 225° = 1.000, which matches 1. ✓
Watch out. Do not forget the sign. Check whether the curve is above or below the axis at that angle.
Method. Use the symmetry of the tan graph: the size of the value is the same as for the related acute angle, and the sign comes from the quarter.
- Find how 315° is related to 45°. Use the cycle of 360°: angles in the first quarter are 0° to 90°, second 90° to 180°, third 180° to 270°, fourth 270° to 360°.315° = 360° − 45° (a fourth-quarter angle)
- Decide whether the value is positive or negative. Look at the graph of tan x: in the fourth quarter the curve is below the x-axis.tan 315° is negative.
- Use the same size with the correct sign. The graph is a mirror image (or turned image) of the first quarter, so the size of the value is the same.tan 315° = −1 = −1
Check. On a calculator, tan 315° = −1.000, which matches −1. ✓
Watch out. Do not forget the sign. Check whether the curve is above or below the axis at that angle.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the sin graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of sin x = 0.5 are the x-values where the horizontal line y = 0.5 crosses the graph of y = sin x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.sin−1(0.5) = 30°
- Use the symmetry of the graph to find the other solutions. The sine graph is symmetrical about x = 90° (and x = 270°). So the second solution in the same cycle is 180° − the first.180 − 30 = 150
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.30, 150
Check. Put each answer into the calculator: sin 30° = 0.50; sin 150° = 0.50 (rounding to 2 d.p. should give about 0.50). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the sin graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of sin x = 0.8 are the x-values where the horizontal line y = 0.8 crosses the graph of y = sin x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.sin−1(0.8) = 53.1°
- Use the symmetry of the graph to find the other solutions. The sine graph is symmetrical about x = 90° (and x = 270°). So the second solution in the same cycle is 180° − the first.180 − 53.1 = 126.9
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.53.1, 126.9
Check. Put each answer into the calculator: sin 53.1° = 0.80; sin 126.9° = 0.80 (rounding to 2 d.p. should give about 0.80). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the sin graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of sin x = −0.5 are the x-values where the horizontal line y = −0.5 crosses the graph of y = sin x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.sin−1(−0.5) = −30°
- Use the symmetry of the graph to find the other solutions. The sine graph is symmetrical about x = 90° (and x = 270°). So the second solution in the same cycle is 180° − the first.180 − (−30) = 210
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 360°. Keep only answers from 0° to 360°.210, 330
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.210, 330
Check. Put each answer into the calculator: sin 210° = −0.50; sin 330° = −0.50 (rounding to 2 d.p. should give about −0.50). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the sin graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of sin x = −0.9 are the x-values where the horizontal line y = −0.9 crosses the graph of y = sin x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.sin−1(−0.9) = −64.2°
- Use the symmetry of the graph to find the other solutions. The sine graph is symmetrical about x = 90° (and x = 270°). So the second solution in the same cycle is 180° − the first.180 − (−64.2) = 244.2
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 360°. Keep only answers from 0° to 360°.244.2, 295.8
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.244.2, 295.8
Check. Put each answer into the calculator: sin 244.2° = −0.90; sin 295.8° = −0.90 (rounding to 2 d.p. should give about −0.90). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the cos graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of cos x = 0.5 are the x-values where the horizontal line y = 0.5 crosses the graph of y = cos x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.cos−1(0.5) = 60°
- Use the symmetry of the graph to find the other solutions. The cosine graph is symmetrical about x = 0° and x = 180°. So the second solution is the negative of the first, then add 360° to bring it into the range.−60 + 360 = 300
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 360°. Keep only answers from 0° to 360°.60, 300
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.60, 300
Check. Put each answer into the calculator: cos 60° = 0.50; cos 300° = 0.50 (rounding to 2 d.p. should give about 0.50). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the cos graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of cos x = −0.5 are the x-values where the horizontal line y = −0.5 crosses the graph of y = cos x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.cos−1(−0.5) = 120°
- Use the symmetry of the graph to find the other solutions. The cosine graph is symmetrical about x = 0° and x = 180°. So the second solution is the negative of the first, then add 360° to bring it into the range.−120 + 360 = 240
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 360°. Keep only answers from 0° to 360°.120, 240
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.120, 240
Check. Put each answer into the calculator: cos 120° = −0.50; cos 240° = −0.50 (rounding to 2 d.p. should give about −0.50). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the cos graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of cos x = 0.2 are the x-values where the horizontal line y = 0.2 crosses the graph of y = cos x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.cos−1(0.2) = 78.5°
- Use the symmetry of the graph to find the other solutions. The cosine graph is symmetrical about x = 0° and x = 180°. So the second solution is the negative of the first, then add 360° to bring it into the range.−78.5 + 360 = 281.5
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 360°. Keep only answers from 0° to 360°.78.5, 281.5
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.78.5, 281.5
Check. Put each answer into the calculator: cos 78.5° = 0.20; cos 281.5° = 0.20 (rounding to 2 d.p. should give about 0.20). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the cos graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of cos x = −0.7 are the x-values where the horizontal line y = −0.7 crosses the graph of y = cos x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.cos−1(−0.7) = 134.4°
- Use the symmetry of the graph to find the other solutions. The cosine graph is symmetrical about x = 0° and x = 180°. So the second solution is the negative of the first, then add 360° to bring it into the range.−134.4 + 360 = 225.6
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 360°. Keep only answers from 0° to 360°.134.4, 225.6
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.134.4, 225.6
Check. Put each answer into the calculator: cos 134.4° = −0.70; cos 225.6° = −0.70 (rounding to 2 d.p. should give about −0.70). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the tan graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of tan x = 1 are the x-values where the horizontal line y = 1 crosses the graph of y = tan x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.tan−1(1) = 45°
- Use the symmetry of the graph to find the other solutions. The tangent graph repeats every 180°, so the other solutions are found by adding (or subtracting) 180°.45 + 180 = 225
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 180°. Keep only answers from 0° to 360°.45, 225
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.45, 225
Check. Put each answer into the calculator: tan 45° = 1.00; tan 225° = 1.00 (rounding to 2 d.p. should give about 1.00). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the tan graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of tan x = −1 are the x-values where the horizontal line y = −1 crosses the graph of y = tan x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.tan−1(−1) = −45°
- Use the symmetry of the graph to find the other solutions. The tangent graph repeats every 180°, so the other solutions are found by adding (or subtracting) 180°.−45 + 180 = 135
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 180°. Keep only answers from 0° to 360°.135, 315
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.135, 315
Check. Put each answer into the calculator: tan 135° = −1.00; tan 315° = −1.00 (rounding to 2 d.p. should give about −1.00). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the tan graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of tan x = 3 are the x-values where the horizontal line y = 3 crosses the graph of y = tan x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.tan−1(3) = 71.6°
- Use the symmetry of the graph to find the other solutions. The tangent graph repeats every 180°, so the other solutions are found by adding (or subtracting) 180°.71.6 + 180 = 251.6
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 180°. Keep only answers from 0° to 360°.71.6, 251.6
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.71.6, 251.6
Check. Put each answer into the calculator: tan 71.6° = 3.00; tan 251.6° = 3.00 (rounding to 2 d.p. should give about 3.00). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the tan graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of tan x = −0.5 are the x-values where the horizontal line y = −0.5 crosses the graph of y = tan x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.tan−1(−0.5) = −26.6°
- Use the symmetry of the graph to find the other solutions. The tangent graph repeats every 180°, so the other solutions are found by adding (or subtracting) 180°.−26.6 + 180 = 153.4
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 180°. Keep only answers from 0° to 360°.153.4, 333.4
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.153.4, 333.4
Check. Put each answer into the calculator: tan 153.4° = −0.50; tan 333.4° = −0.50 (rounding to 2 d.p. should give about −0.50). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the sin graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of sin x = 0.5 are the x-values where the horizontal line y = 0.5 crosses the graph of y = sin x. Between -180° and 180° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.sin−1(0.5) = 30°
- Use the symmetry of the graph to find the other solutions. The sine graph is symmetrical about x = 90° (and x = 270°). So the second solution in the same cycle is 180° − the first.180 − 30 = 150
- Write the solutions. There are 2 solutions in the range -180° ≤ x ≤ 180°.30, 150
Check. Put each answer into the calculator: sin 30° = 0.50; sin 150° = 0.50 (rounding to 2 d.p. should give about 0.50). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the cos graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of cos x = 0.5 are the x-values where the horizontal line y = 0.5 crosses the graph of y = cos x. Between -180° and 180° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.cos−1(0.5) = 60°
- Use the symmetry of the graph to find the other solutions. The cosine graph is symmetrical about x = 0° and x = 180°. So the second solution is the negative of the first, then add 360° to bring it into the range.−60 + 360 = 300
- Write the solutions. There are 2 solutions in the range -180° ≤ x ≤ 180°.−60, 60
Check. Put each answer into the calculator: cos −60° = 0.50; cos 60° = 0.50 (rounding to 2 d.p. should give about 0.50). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the tan graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of tan x = 1 are the x-values where the horizontal line y = 1 crosses the graph of y = tan x. Between -180° and 180° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.tan−1(1) = 45°
- Use the symmetry of the graph to find the other solutions. The tangent graph repeats every 180°, so the other solutions are found by adding (or subtracting) 180°.45 + 180 = 225
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 180°. Keep only answers from -180° to 180°.−135, 45
- Write the solutions. There are 2 solutions in the range -180° ≤ x ≤ 180°.−135, 45
Check. Put each answer into the calculator: tan −135° = 1.00; tan 45° = 1.00 (rounding to 2 d.p. should give about 1.00). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the sin graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of sin x = −0.4 are the x-values where the horizontal line y = −0.4 crosses the graph of y = sin x. Between -180° and 180° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.sin−1(−0.4) = −23.6°
- Use the symmetry of the graph to find the other solutions. The sine graph is symmetrical about x = 90° (and x = 270°). So the second solution in the same cycle is 180° − the first.180 − (−23.6) = 203.6
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 360°. Keep only answers from -180° to 180°.−156.4, −23.6
- Write the solutions. There are 2 solutions in the range -180° ≤ x ≤ 180°.−156.4, −23.6
Check. Put each answer into the calculator: sin −156.4° = −0.40; sin −23.6° = −0.40 (rounding to 2 d.p. should give about −0.40). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Key words. The period of a graph is the length of one complete cycle before it repeats. sin and cos repeat every 360°; tan repeats every 180°.
Method. Find the first solution with the inverse function, then use the symmetry and the repeat pattern of the sin graph to find the rest in the range.
- Write the equation and picture the graph. The solutions of sin x = 0.3 are the x-values where the horizontal line y = 0.3 crosses the graph of y = sin x. Between 0° and 720° it crosses 4 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.sin−1(0.3) = 17.5°
- Use the symmetry of the graph to find the other solutions. The sine graph is symmetrical about x = 90° (and x = 270°). So the second solution in the same cycle is 180° − the first.180 − 17.5 = 162.5
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 360°. Keep only answers from 0° to 720°.17.5, 162.5, 377.5, 522.5
- Write the solutions. There are 4 solutions in the range 0° ≤ x ≤ 720°.17.5, 162.5, 377.5, 522.5
Check. Put each answer into the calculator: sin 17.5° = 0.30; sin 162.5° = 0.30; sin 377.5° = 0.30; sin 522.5° = 0.30 (rounding to 2 d.p. should give about 0.30). ✓
Watch out. The calculator only gives one answer. Always ask yourself where the graph crosses the line again.
Method. Read the amplitude, the period and the middle line from the graph, then match them to y = a sin(bx) + d (or cos).
- Read the maximum and minimum values. The graph oscillates between a highest and lowest value.maximum = 3, minimum = −3
- Find the amplitude and the middle line. The amplitude is half the distance between the maximum and the minimum. The middle line is halfway between them.amplitude = (3 − (−3)) ÷ 2 = 3
middle line: y = (3 + (−3)) ÷ 2 = 0 - Read the period (the length of one full cycle). The standard sin graph has period 360°. If one cycle takes 360° then the angle is multiplied by 1.period = 360° ⇒ b = 360 ÷ 360 = 1
- Write the equation. Use y = a sin (bx) + d with a = 3, b = 1, d = 0.y = 3 sin x
Check. Test one point: when x = 0, the equation gives 0.00, and the graph passes through that value. ✓
Watch out. The period is the length of one whole cycle, not half of it.
Method. Read the amplitude, the period and the middle line from the graph, then match them to y = a sin(bx) + d (or cos).
- Read the maximum and minimum values. The graph oscillates between a highest and lowest value.maximum = 1, minimum = −1
- Find the amplitude and the middle line. The amplitude is half the distance between the maximum and the minimum. The middle line is halfway between them.amplitude = (1 − (−1)) ÷ 2 = 1
middle line: y = (1 + (−1)) ÷ 2 = 0 - Read the period (the length of one full cycle). The standard cos graph has period 360°. If one cycle takes 180° then the angle is multiplied by 2.period = 180° ⇒ b = 360 ÷ 180 = 2
- Write the equation. Use y = a cos (bx) + d with a = 1, b = 2, d = 0.y = cos 2x
Check. Test one point: when x = 0, the equation gives 1.00, and the graph passes through that value. ✓
Watch out. The period is the length of one whole cycle, not half of it.
Method. Read the amplitude, the period and the middle line from the graph, then match them to y = a sin(bx) + d (or cos).
- Read the maximum and minimum values. The graph oscillates between a highest and lowest value.maximum = 3, minimum = −1
- Find the amplitude and the middle line. The amplitude is half the distance between the maximum and the minimum. The middle line is halfway between them.amplitude = (3 − (−1)) ÷ 2 = 2
middle line: y = (3 + (−1)) ÷ 2 = 1 - Read the period (the length of one full cycle). The standard sin graph has period 360°. If one cycle takes 360° then the angle is multiplied by 1.period = 360° ⇒ b = 360 ÷ 360 = 1
- Write the equation. Use y = a sin (bx) + d with a = 2, b = 1, d = 1.y = 2 sin x + 1
Check. Test one point: when x = 0, the equation gives 1.00, and the graph passes through that value. ✓
Watch out. The period is the length of one whole cycle, not half of it.
Method. For y = a sin(bx) + d or y = a cos(bx) + d: the maximum is d + a, the minimum is d − a, and the period is 360° ÷ b.
- Find the maximum and minimum. The amplitude a is how far the wave goes above and below its middle line y = d.maximum = 0 + 2 = 2
minimum = 0 − 2 = −2 - Find the period. The pattern repeats after 360° divided by the number multiplying x.period = 360° ÷ 1 = 360°
- Find the y-intercept. Put x = 0 into the equation.y = 2 cos 0° = 2
Check. The maximum and minimum are 2 and −2, so the wave is centred on 0. ✓
Watch out. The amplitude is half the distance between the maximum and minimum, not the whole distance.
Method. For y = a sin(bx) + d or y = a cos(bx) + d: the maximum is d + a, the minimum is d − a, and the period is 360° ÷ b.
- Find the maximum and minimum. The amplitude a is how far the wave goes above and below its middle line y = d.maximum = 1 + 1 = 2
minimum = 1 − 1 = 0 - Find the period. The pattern repeats after 360° divided by the number multiplying x.period = 360° ÷ 3 = 120°
- Find the y-intercept. Put x = 0 into the equation.y = sin 0° + 1 = 1
Check. The maximum and minimum are 2 and 0, so the wave is centred on 1. ✓
Watch out. The amplitude is half the distance between the maximum and minimum, not the whole distance.
Method. For y = a sin(bx) + d or y = a cos(bx) + d: the maximum is d + a, the minimum is d − a, and the period is 360° ÷ b.
- Find the maximum and minimum. The amplitude a is how far the wave goes above and below its middle line y = d.maximum = −1 + 4 = 3
minimum = −1 − 4 = −5 - Find the period. The pattern repeats after 360° divided by the number multiplying x.period = 360° ÷ 2 = 180°
- Find the y-intercept. Put x = 0 into the equation.y = 4 cos 0° − 1 = 3
Check. The maximum and minimum are 3 and −5, so the wave is centred on −1. ✓
Watch out. The amplitude is half the distance between the maximum and minimum, not the whole distance.
Method. Substitute the times into the formula. Remember 30t is the angle in degrees.
- At t = 3. The angle is 30 × 3 = 90°.h = 5 + 3 sin 90° = 5 + 3 × 1 = 8
- At t = 9. The angle is 30 × 9 = 270°.h = 5 + 3 sin 270° = 5 + 3 × (−1) = 2
Check. High tide is 8 m, low tide 2 m. ✓
Watch out. Use the angle 30t, not t, in the sine.
Method. The sine part goes between −1 and 1.
- Greatest depth. The biggest value of sin is 1.5 + 3 × 1 = 8
- Least depth. The smallest value of sin is −1.5 + 3 × (−1) = 2
Check. The depth varies by 3 m either side of 5 m. ✓
Watch out. Put −1 in brackets.
Method. Set the depth equal to 6.5 and solve for t using the graph of sine.
- Write the equation. 5 + 3 sin(30t)° = 6.5
- Rearrange to get sin on its own. Subtract 5, then divide by 3.3 sin(30t)° = 1.5, so sin(30t)° = 0.5
- Solve for the angle. sin 30° = 0.5, and the other solution in the cycle is 180° − 30° = 150°.30t = 30 or 30t = 150
- Divide by 30. t = 1 or t = 5
Check. At t = 1: 5 + 3 sin 30° = 5 + 1.5 = 6.5. At t = 5: 5 + 3 sin 150° = 6.5. ✓
Watch out. The range 0 ≤ t ≤ 12 means the angle 30t goes up to 360°, so there are two solutions.
Method. Use the translation rule f(x − 90) and compare values from a table.
- Describe the translation. Subtracting 90 inside the bracket moves the graph 90 to the right.translation by the vector (90°, 0)
- Work out the values. For each x, find x − 90 then take the cosine.
x x − 90° cos(x − 90°) sin x 0° -90° 0.00 0.00 30° -60° 0.50 0.50 90° 0° 1.00 1.00 150° 60° 0.50 0.50 - Compare. The two columns are equal in every row.So cos(x − 90°) = sin x
Check. The cosine graph starts at its maximum at x = 0. Moving it 90° right puts the maximum at 90°, which is where sin x has its maximum. ✓
Watch out. A translation to the right means subtracting inside the bracket.
Method. Think about the range of values each graph covers.
- Look at the range of sin x. The sine graph never goes above 1 or below −1.−1 ≤ sin x ≤ 1
- Compare with 2. 2 is bigger than the maximum value of sin x, so the horizontal line y = 2 never meets the graph.sin x = 2 has no solutions.
- Look at the range of tan x. The tangent graph rises without limit near each asymptote, so it takes every value, however large.tan x = 2000 has solutions (about 89.97° and 269.97°).
Check. Calculator: tan−1(2000) = 89.97°. And sin−1(2) gives a maths error. ✓
Watch out. The calculator showing an error for sin−1(2) is telling you there is no solution.
Method. First rearrange to get sin x on its own. Then solve it using the graph and the inverse button.
- Write the equation. 2 sin x + 1 = 0
- Subtract 1 from both sides. We want sin x on its own, so undo the + 1 first.2 sin x = −1
- Divide both sides by 2. The sin is multiplied by 2.sin x = −1 ÷ 2 = −0.5
- Write the equation and picture the graph. The solutions of sin x = −0.5 are the x-values where the horizontal line y = −0.5 crosses the graph of y = sin x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.sin−1(−0.5) = −30°
- Use the symmetry of the graph to find the other solutions. The sine graph is symmetrical about x = 90° (and x = 270°). So the second solution in the same cycle is 180° − the first.180 − (−30) = 210
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 360°. Keep only answers from 0° to 360°.210, 330
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.210, 330
Check. Put the first answer into the original equation: 2 sin x + 1 with x = 210° gives 0.00, which is 0 to 2 d.p. ✓
Watch out. Rearrange before using the inverse button. Do not put the 2 or the +1 into the calculator with the angle.
Method. First rearrange to get cos x on its own. Then solve it using the graph and the inverse button.
- Write the equation. 2 cos x + 1 = 0.2
- Subtract 1 from both sides. We want cos x on its own, so undo the + 1 first.2 cos x = −0.8
- Divide both sides by 2. The cos is multiplied by 2.cos x = −0.8 ÷ 2 = −0.4
- Write the equation and picture the graph. The solutions of cos x = −0.4 are the x-values where the horizontal line y = −0.4 crosses the graph of y = cos x. Between 0° and 360° it crosses 2 times.
- Use the inverse button on the calculator for the first (principal) solution. The calculator gives one answer, which may not be in the range we want.cos−1(−0.4) = 113.6°
- Use the symmetry of the graph to find the other solutions. The cosine graph is symmetrical about x = 0° and x = 180°. So the second solution is the negative of the first, then add 360° to bring it into the range.−113.6 + 360 = 246.4
- Add or subtract whole cycles until every answer is in the range. The graph repeats every 360°. Keep only answers from 0° to 360°.113.6, 246.4
- Write the solutions. There are 2 solutions in the range 0° ≤ x ≤ 360°.113.6, 246.4
Check. Put the first answer into the original equation: 2 cos x + 1 with x = 113.6° gives 0.20, which is 0.2 to 2 d.p. ✓
Watch out. Rearrange before using the inverse button. Do not put the 2 or the +1 into the calculator with the angle.
Method. At a crossing point sin x = cos x. Divide both sides by cos x and use tan x = sin x ÷ cos x.
- Write what happens at a crossing point. The two curves have the same height there.sin x = cos x
- Divide both sides by cos x. This is allowed because cos x is not 0 at these points (if it were, sin x would be ±1, not 0).sin x ÷ cos x = 1
- Use tan x = sin x ÷ cos x. tan x = 1
- Solve for 0° ≤ x ≤ 360°. The calculator gives 45°. The tangent graph repeats every 180°, so add 180°.x = 45° or 45° + 180° = 225°
Check. sin 45° = cos 45° = 0.707 and sin 225° = cos 225° = −0.707. ✓
Watch out. Do not divide by sin x or cos x without checking that it is not zero.