GCSE Maths · Pythagoras and Trigonometry · Foundation · Summary
GCSE Maths Pythagoras and Trigonometry (Foundation): summary and key words
Part of the GCSE Maths pythagoras and trigonometry chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page
Chapter summary
| Skill | Method |
|---|---|
| Hypotenuse (Pythagoras) | Add the squares of the two shorter sides, then square root: c = √(a2 + b2) |
| Shorter side (Pythagoras) | Subtract the squares: a = √(c2 − b2) |
| Is a triangle right-angled? | Does (short)2 + (short)2 = (longest)2? If equal, yes |
| Distance between two points | Horizontal and vertical steps are the shorter sides; d2 = (x step)2 + (y step)2 |
| Isosceles and equilateral triangles | Split down the middle: half the base, the height, the slanting side make a right-angled triangle |
| Label the sides | Hypotenuse is opposite the right angle; opposite is across from θ; adjacent is next to θ |
| SOH CAH TOA | sin θ = O÷H, cos θ = A÷H, tan θ = O÷A |
| Find a side with trigonometry | Label, choose the ratio, write the equation, rearrange (unknown on top: multiply, on the bottom: divide), calculate (degree mode) |
| Find an angle with trigonometry | Write the ratio as a fraction, then use the inverse: θ = sin−1, cos−1 or tan−1 |
| Elevation and depression | Measured from the horizontal; the angle of depression from the top equals the angle of elevation from the bottom (alternate angles) |
| Bearings | Clockwise from north, three figures; back bearing = bearing ± 180°; east = d sin b, north = d cos b |
| Exact values | sin 30° = ½, sin 45° = √2÷2, sin 60° = √3÷2; cos is the reverse; tan 30° = √3÷3, tan 45° = 1, tan 60° = √3 |
| Sine rule (Higher) | a÷sin A = b÷sin B = c÷sin C; needs a matching pair; check for the obtuse angle |
| Cosine rule (Higher) | a2 = b2 + c2 − 2bc cos A; for SAS and SSS; cos A = (b2 + c2 − a2) ÷ 2bc |
| Area of a triangle (Higher) | Area = ½ ab sin C (C is between sides a and b) |
| 3D problems (Higher) | Find right-angled triangles, draw them flat; angle between a line and a plane = angle with its shadow |
Can you…?
- I can use Pythagoras’ theorem to find the hypotenuse and a shorter side.
- I can decide whether a triangle is right-angled and recognise Pythagorean triples.
- I can solve problems with coordinates and with isosceles and equilateral triangles using Pythagoras.
- I can label the opposite, adjacent and hypotenuse from any angle.
- I can use sin, cos and tan to find a side, including when the unknown is on the bottom.
- I can use sin−1, cos−1 and tan−1 to find an angle.
- I can solve problems with angles of elevation and depression and with bearings.
- I know the exact values of sin, cos and tan for 0°, 30°, 45°, 60° and 90°.
- Higher: I can use the sine rule, including the ambiguous case.
- Higher: I can use the cosine rule to find a side or an angle.
- Higher: I can find the area of a triangle with ½ ab sin C.
- Higher: I can solve 3D problems and problems that combine several rules.
Key terms
- Hypotenuse
- The longest side of a right-angled triangle, opposite the right angle.
- Pythagoras’ theorem
- In a right-angled triangle, a2 + b2 = c2, where c is the hypotenuse.
- Pythagorean triple
- Three whole numbers a, b, c with a2 + b2 = c2, such as 3, 4, 5.
- Opposite
- The side across from the angle θ, not touching it.
- Adjacent
- The side next to the angle θ that is not the hypotenuse.
- Sine, cosine, tangent
- The ratios O÷H, A÷H and O÷A of a right-angled triangle.
- Inverse function
- sin−1, cos−1, tan−1: turns a ratio back into an angle.
- Angle of elevation
- The angle upwards from the horizontal to a line of sight.
- Angle of depression
- The angle downwards from the horizontal to a line of sight.
- Bearing
- A direction measured clockwise from north, written with three figures.
- Exact value
- An answer left as a fraction or surd, not as a rounded decimal.
- Rationalise
- Remove a square root from the bottom of a fraction by multiplying top and bottom by it.
- Sine rule
- a÷sin A = b÷sin B = c÷sin C (Higher).
- Cosine rule
- a2 = b2 + c2 − 2bc cos A (Higher).
- Included angle
- The angle between two given sides.
- Space diagonal
- A line joining opposite corners of a cuboid through the inside of the solid.
- Projection (shadow)
- The shadow of a line on a plane, used to find the angle between a line and a plane.