GCSE Maths: The cosine rule (Higher)
Part of the GCSE Maths pythagoras and trigonometry chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
The cosine rule
The sine rule needs a matching pair (a side and its opposite angle). What if you do not have one? For example, if you know two sides and the angle between them, or all three sides. Then you use the cosine rule.
a2 = b2 + c2 − 2bc cos A
Here a is the side opposite angle A, and b and c are the two other sides. The angle A is the angle between sides b and c. It looks like Pythagoras’ theorem with a correction term. If A = 90° then cos A = 0, the last term disappears and we get a2 = b2 + c2. If A is bigger than 90°, cos A is negative, so −2bc cos A is positive and a is longer than Pythagoras would give. That makes sense: a wider angle gives a longer opposite side.
Finding a side (two sides and the angle between them, SAS). Substitute the numbers, work out 2bc cos A first as one number, subtract it from b2 + c2, and take the square root.
Finding an angle (three sides, SSS). Rearrange the rule to make cos A the subject:
cos A = b2 + c2 − a22bc
Then use the inverse cosine. If the top is negative, the angle is obtuse, and the calculator gives it correctly (unlike the sine rule, there is no second answer).
Which rule? Look at what is given. A matching side and angle pair, plus one more value: sine rule. Two sides and the angle between them (SAS), or three sides (SSS): cosine rule. Start with the cosine rule when you cannot find a matching pair.
Key ideas
- Cosine rule: a2 = b2 + c2 − 2bc cos A.
- To find an angle: cos A = (b2 + c2 − a2) ÷ 2bc.
- Use it for SAS (two sides and the angle between) and SSS (three sides).
- Work out 2bc cos A before subtracting.
- cos of an obtuse angle is negative.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. The cosine rule finds the third side of a triangle when two sides and the angle between them are known: a2 = b2 + c2 − 2bc cos A.
- Spot the angle and the two sides around it. The cosine rule is used when we know two sides and the angle between them. The side we want is the one opposite that angle.Known: b = 7 cm, c = 9 cm, angle A = 50° between them.
Wanted: a, opposite angle A. - Write the cosine rule and put the numbers in. Square the wanted side on the left; square the other two sides on the right, then take away 2 × side × side × cos(angle).a2 = b2 + c2 − 2bc cos A
a2 = 72 + 92 − 2 × 7 × 9 × cos 50° - Work out the two squares and add. This is the first part, b2 + c2.72 + 92 = 49 + 81 = 130
- Work out the last part on its own. Multiply everything together first. Do this before subtracting (BIDMAS: multiplication comes before subtraction).cos 50° = 0.64278…
2 × 7 × 9 × 0.64278… = 80.99123… - Subtract. This gives a2.a2 = 130 − 80.99123… = 49.00876…
- Square root, then round to 1 d.p.. We want a, not a2.a = √49.00876… = 7.00062…
a = 7.0 cm
Check. One side of a triangle is between the difference and the sum of the other two: 2 is less than 7.0, which is less than 16. Angle A is less than 90°, so a2 should be less than b2 + c2: 49.00876… is less than 130. ✓
Watch out. Do 2bc cos A first and then subtract it. A very common mistake is to work out (b2 + c2 − 2bc) first. The rule has a minus sign, even for an obtuse angle (cos of an obtuse angle is negative).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. The cosine rule finds the third side of a triangle when two sides and the angle between them are known: a2 = b2 + c2 − 2bc cos A.
- Spot the angle and the two sides around it. The cosine rule is used when we know two sides and the angle between them. The side we want is the one opposite that angle.Known: b = 6 cm, c = 8 cm, angle A = 120° between them.
Wanted: a, opposite angle A. - Write the cosine rule and put the numbers in. Square the wanted side on the left; square the other two sides on the right, then take away 2 × side × side × cos(angle).a2 = b2 + c2 − 2bc cos A
a2 = 62 + 82 − 2 × 6 × 8 × cos 120° - Work out the two squares and add. This is the first part, b2 + c2.62 + 82 = 36 + 64 = 100
- Work out the last part on its own. Multiply everything together first. Do this before subtracting (BIDMAS: multiplication comes before subtraction).cos 120° = −0.5
2 × 6 × 8 × −0.5 = −48 - Subtract. This gives a2.a2 = 100 − −48 = 148
- Square root, then round to 1 d.p.. We want a, not a2.a = √148 = 12.16552…
a = 12.2 cm
Check. One side of a triangle is between the difference and the sum of the other two: 2 is less than 12.2, which is less than 14. Angle A is more than 90°, so a2 should be more than b2 + c2: 148 is more than 100. ✓
Watch out. Do 2bc cos A first and then subtract it. A very common mistake is to work out (b2 + c2 − 2bc) first. The rule has a minus sign, even for an obtuse angle (cos of an obtuse angle is negative).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. When all three sides are known, rearrange the cosine rule to cos A = (b2 + c2 − a2) ÷ 2bc, then use the inverse cosine.
- Spot the angle we want and the side opposite it. With three sides known we can find any angle. Call the side opposite the wanted angle a.Wanted: angle A, opposite the side a = 11 cm.
The other two sides are b = 7 cm and c = 9 cm. - Write the cosine rule and rearrange it for cos A. We want cos A on its own, so add 2bc cos A to both sides, subtract a2 from both sides, then divide by 2bc.a2 = b2 + c2 − 2bc cos A
2bc cos A = b2 + c2 − a2
cos A = b2 + c2 − a22bc - Put the numbers in the top and the bottom. Work the top and the bottom out separately so you do not make a slip.Top: 72 + 92 − 112 = 49 + 81 − 121 = 9
Bottom: 2 × 7 × 9 = 126 - Divide to find cos A. This gives the cosine of the angle as a decimal.cos A = 9126 = 0.07142…
- Use the inverse cosine. The inverse button (cos−1) turns the cosine value back into an angle. (A negative cosine gives an obtuse angle, which is correct.)A = cos−1(0.07142…) = 85.90395…°
- Round to 1 d.p. Give the angle to the accuracy asked.A = 85.9°
Check. Put the angle back in the cosine rule: 72 + 92 − 2 × 7 × 9 × cos 85.9° = 121.0, and 112 = 121.0. ✓
Watch out. In the top line the side a opposite the wanted angle is the one that is subtracted. Work out the top and the bottom separately, and do not divide until both are finished.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. When all three sides are known, rearrange the cosine rule to cos A = (b2 + c2 − a2) ÷ 2bc, then use the inverse cosine.
- Spot the angle we want and the side opposite it. With three sides known we can find any angle. Call the side opposite the wanted angle a.Wanted: angle A, opposite the side a = 10 cm.
The other two sides are b = 5 cm and c = 6 cm. - Write the cosine rule and rearrange it for cos A. We want cos A on its own, so add 2bc cos A to both sides, subtract a2 from both sides, then divide by 2bc.a2 = b2 + c2 − 2bc cos A
2bc cos A = b2 + c2 − a2
cos A = b2 + c2 − a22bc - Put the numbers in the top and the bottom. Work the top and the bottom out separately so you do not make a slip.Top: 52 + 62 − 102 = 25 + 36 − 100 = −39
Bottom: 2 × 5 × 6 = 60 - Divide to find cos A. This gives the cosine of the angle as a decimal.cos A = −3960 = −0.65
- Use the inverse cosine. The inverse button (cos−1) turns the cosine value back into an angle. (A negative cosine gives an obtuse angle, which is correct.)A = cos−1(−0.65) = 130.54160…°
- Round to 1 d.p. Give the angle to the accuracy asked.A = 130.5°
Check. Put the angle back in the cosine rule: 52 + 62 − 2 × 5 × 6 × cos 130.5° = 100.0, and 102 = 100.0. ✓
Watch out. In the top line the side a opposite the wanted angle is the one that is subtracted. Work out the top and the bottom separately, and do not divide until both are finished.
Watch out
Do the multiplication 2bc cos A first. Do not work out (b2 + c2 − 2bc) first and then multiply by cos A. And the angle must be the one between the two sides that are squared and added. If you find an obtuse angle, cos A is negative, so be careful with the signs.Exercise 20K The cosine rule
Calculator allowedKey words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. The cosine rule finds the third side of a triangle when two sides and the angle between them are known: a2 = b2 + c2 − 2bc cos A.
- Spot the angle and the two sides around it. The cosine rule is used when we know two sides and the angle between them. The side we want is the one opposite that angle.Known: b = 8 cm, c = 5 cm, angle A = 60° between them.
Wanted: a, opposite angle A. - Write the cosine rule and put the numbers in. Square the wanted side on the left; square the other two sides on the right, then take away 2 × side × side × cos(angle).a2 = b2 + c2 − 2bc cos A
a2 = 82 + 52 − 2 × 8 × 5 × cos 60° - Work out the two squares and add. This is the first part, b2 + c2.82 + 52 = 64 + 25 = 89
- Work out the last part on its own. Multiply everything together first. Do this before subtracting (BIDMAS: multiplication comes before subtraction).cos 60° = 0.5
2 × 8 × 5 × 0.5 = 40 - Subtract. This gives a2.a2 = 89 − 40 = 49
- Square root, then round to 1 d.p.. We want a, not a2.a = √49 = 7
a = 7.0 cm
Check. One side of a triangle is between the difference and the sum of the other two: 3 is less than 7.0, which is less than 13. Angle A is less than 90°, so a2 should be less than b2 + c2: 49 is less than 89. ✓
Watch out. Do 2bc cos A first and then subtract it. A very common mistake is to work out (b2 + c2 − 2bc) first. The rule has a minus sign, even for an obtuse angle (cos of an obtuse angle is negative).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. The cosine rule finds the third side of a triangle when two sides and the angle between them are known: a2 = b2 + c2 − 2bc cos A.
- Spot the angle and the two sides around it. The cosine rule is used when we know two sides and the angle between them. The side we want is the one opposite that angle.Known: b = 9.5 cm, c = 6.2 cm, angle A = 35° between them.
Wanted: a, opposite angle A. - Write the cosine rule and put the numbers in. Square the wanted side on the left; square the other two sides on the right, then take away 2 × side × side × cos(angle).a2 = b2 + c2 − 2bc cos A
a2 = 9.52 + 6.22 − 2 × 9.5 × 6.2 × cos 35° - Work out the two squares and add. This is the first part, b2 + c2.9.52 + 6.22 = 90.25 + 38.44 = 128.69
- Work out the last part on its own. Multiply everything together first. Do this before subtracting (BIDMAS: multiplication comes before subtraction).cos 35° = 0.81915…
2 × 9.5 × 6.2 × 0.81915… = 96.49611… - Subtract. This gives a2.a2 = 128.69 − 96.49611… = 32.19388…
- Square root, then round to 1 d.p.. We want a, not a2.a = √32.19388… = 5.67396…
a = 5.7 cm
Check. One side of a triangle is between the difference and the sum of the other two: 3.3 is less than 5.7, which is less than 15.7. Angle A is less than 90°, so a2 should be less than b2 + c2: 32.19388… is less than 128.69. ✓
Watch out. Do 2bc cos A first and then subtract it. A very common mistake is to work out (b2 + c2 − 2bc) first. The rule has a minus sign, even for an obtuse angle (cos of an obtuse angle is negative).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. The cosine rule finds the third side of a triangle when two sides and the angle between them are known: a2 = b2 + c2 − 2bc cos A.
- Spot the angle and the two sides around it. The cosine rule is used when we know two sides and the angle between them. The side we want is the one opposite that angle.Known: b = 12 cm, c = 7 cm, angle A = 72° between them.
Wanted: a, opposite angle A. - Write the cosine rule and put the numbers in. Square the wanted side on the left; square the other two sides on the right, then take away 2 × side × side × cos(angle).a2 = b2 + c2 − 2bc cos A
a2 = 122 + 72 − 2 × 12 × 7 × cos 72° - Work out the two squares and add. This is the first part, b2 + c2.122 + 72 = 144 + 49 = 193
- Work out the last part on its own. Multiply everything together first. Do this before subtracting (BIDMAS: multiplication comes before subtraction).cos 72° = 0.30901…
2 × 12 × 7 × 0.30901… = 51.91485… - Subtract. This gives a2.a2 = 193 − 51.91485… = 141.08514…
- Square root, then round to 1 d.p.. We want a, not a2.a = √141.08514… = 11.87792…
a = 11.9 cm
Check. One side of a triangle is between the difference and the sum of the other two: 5 is less than 11.9, which is less than 19. Angle A is less than 90°, so a2 should be less than b2 + c2: 141.08514… is less than 193. ✓
Watch out. Do 2bc cos A first and then subtract it. A very common mistake is to work out (b2 + c2 − 2bc) first. The rule has a minus sign, even for an obtuse angle (cos of an obtuse angle is negative).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. The cosine rule finds the third side of a triangle when two sides and the angle between them are known: a2 = b2 + c2 − 2bc cos A.
- Spot the angle and the two sides around it. The cosine rule is used when we know two sides and the angle between them. The side we want is the one opposite that angle.Known: b = 4.3 cm, c = 6.8 cm, angle A = 48° between them.
Wanted: a, opposite angle A. - Write the cosine rule and put the numbers in. Square the wanted side on the left; square the other two sides on the right, then take away 2 × side × side × cos(angle).a2 = b2 + c2 − 2bc cos A
a2 = 4.32 + 6.82 − 2 × 4.3 × 6.8 × cos 48° - Work out the two squares and add. This is the first part, b2 + c2.4.32 + 6.82 = 18.49 + 46.24 = 64.72999999999999
- Work out the last part on its own. Multiply everything together first. Do this before subtracting (BIDMAS: multiplication comes before subtraction).cos 48° = 0.66913…
2 × 4.3 × 6.8 × 0.66913… = 39.13075… - Subtract. This gives a2.a2 = 64.72999999999999 − 39.13075… = 25.59924…
- Square root, then round to 1 d.p.. We want a, not a2.a = √25.59924… = 5.05956…
a = 5.1 cm
Check. One side of a triangle is between the difference and the sum of the other two: 2.5 is less than 5.1, which is less than 11.1. Angle A is less than 90°, so a2 should be less than b2 + c2: 25.59924… is less than 64.72999999999999. ✓
Watch out. Do 2bc cos A first and then subtract it. A very common mistake is to work out (b2 + c2 − 2bc) first. The rule has a minus sign, even for an obtuse angle (cos of an obtuse angle is negative).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. The cosine rule finds the third side of a triangle when two sides and the angle between them are known: a2 = b2 + c2 − 2bc cos A.
- Spot the angle and the two sides around it. The cosine rule is used when we know two sides and the angle between them. The side we want is the one opposite that angle.Known: b = 5 cm, c = 7 cm, angle A = 110° between them.
Wanted: a, opposite angle A. - Write the cosine rule and put the numbers in. Square the wanted side on the left; square the other two sides on the right, then take away 2 × side × side × cos(angle).a2 = b2 + c2 − 2bc cos A
a2 = 52 + 72 − 2 × 5 × 7 × cos 110° - Work out the two squares and add. This is the first part, b2 + c2.52 + 72 = 25 + 49 = 74
- Work out the last part on its own. Multiply everything together first. Do this before subtracting (BIDMAS: multiplication comes before subtraction).cos 110° = −0.34202…
2 × 5 × 7 × −0.34202… = −23.94141… - Subtract. This gives a2.a2 = 74 − −23.94141… = 97.94141…
- Square root, then round to 1 d.p.. We want a, not a2.a = √97.94141… = 9.89653…
a = 9.9 cm
Check. One side of a triangle is between the difference and the sum of the other two: 2 is less than 9.9, which is less than 12. Angle A is more than 90°, so a2 should be more than b2 + c2: 97.94141… is more than 74. ✓
Watch out. Do 2bc cos A first and then subtract it. A very common mistake is to work out (b2 + c2 − 2bc) first. The rule has a minus sign, even for an obtuse angle (cos of an obtuse angle is negative).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. The cosine rule finds the third side of a triangle when two sides and the angle between them are known: a2 = b2 + c2 − 2bc cos A.
- Spot the angle and the two sides around it. The cosine rule is used when we know two sides and the angle between them. The side we want is the one opposite that angle.Known: b = 8.2 cm, c = 9.1 cm, angle A = 128° between them.
Wanted: a, opposite angle A. - Write the cosine rule and put the numbers in. Square the wanted side on the left; square the other two sides on the right, then take away 2 × side × side × cos(angle).a2 = b2 + c2 − 2bc cos A
a2 = 8.22 + 9.12 − 2 × 8.2 × 9.1 × cos 128° - Work out the two squares and add. This is the first part, b2 + c2.8.22 + 9.12 = 67.24 + 82.81 = 150.04999999999998
- Work out the last part on its own. Multiply everything together first. Do this before subtracting (BIDMAS: multiplication comes before subtraction).cos 128° = −0.61566…
2 × 8.2 × 9.1 × −0.61566… = −91.88131… - Subtract. This gives a2.a2 = 150.04999999999998 − −91.88131… = 241.93131…
- Square root, then round to 1 d.p.. We want a, not a2.a = √241.93131… = 15.55414…
a = 15.6 cm
Check. One side of a triangle is between the difference and the sum of the other two: 0.9000000000000004 is less than 15.6, which is less than 17.299999999999997. Angle A is more than 90°, so a2 should be more than b2 + c2: 241.93131… is more than 150.04999999999998. ✓
Watch out. Do 2bc cos A first and then subtract it. A very common mistake is to work out (b2 + c2 − 2bc) first. The rule has a minus sign, even for an obtuse angle (cos of an obtuse angle is negative).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. When all three sides are known, rearrange the cosine rule to cos A = (b2 + c2 − a2) ÷ 2bc, then use the inverse cosine.
- Spot the angle we want and the side opposite it. With three sides known we can find any angle. Call the side opposite the wanted angle a.Wanted: angle A, opposite the side a = 7 cm.
The other two sides are b = 8 cm and c = 9 cm. - Write the cosine rule and rearrange it for cos A. We want cos A on its own, so add 2bc cos A to both sides, subtract a2 from both sides, then divide by 2bc.a2 = b2 + c2 − 2bc cos A
2bc cos A = b2 + c2 − a2
cos A = b2 + c2 − a22bc - Put the numbers in the top and the bottom. Work the top and the bottom out separately so you do not make a slip.Top: 82 + 92 − 72 = 64 + 81 − 49 = 96
Bottom: 2 × 8 × 9 = 144 - Divide to find cos A. This gives the cosine of the angle as a decimal.cos A = 96144 = 0.66666…
- Use the inverse cosine. The inverse button (cos−1) turns the cosine value back into an angle. (A negative cosine gives an obtuse angle, which is correct.)A = cos−1(0.66666…) = 48.18968…°
- Round to 1 d.p. Give the angle to the accuracy asked.A = 48.2°
Check. Put the angle back in the cosine rule: 82 + 92 − 2 × 8 × 9 × cos 48.2° = 49.0, and 72 = 49.0. ✓
Watch out. In the top line the side a opposite the wanted angle is the one that is subtracted. Work out the top and the bottom separately, and do not divide until both are finished.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. When all three sides are known, rearrange the cosine rule to cos A = (b2 + c2 − a2) ÷ 2bc, then use the inverse cosine.
- Spot the angle we want and the side opposite it. With three sides known we can find any angle. Call the side opposite the wanted angle a.Wanted: angle A, opposite the side a = 12 cm.
The other two sides are b = 9 cm and c = 10 cm. - Write the cosine rule and rearrange it for cos A. We want cos A on its own, so add 2bc cos A to both sides, subtract a2 from both sides, then divide by 2bc.a2 = b2 + c2 − 2bc cos A
2bc cos A = b2 + c2 − a2
cos A = b2 + c2 − a22bc - Put the numbers in the top and the bottom. Work the top and the bottom out separately so you do not make a slip.Top: 92 + 102 − 122 = 81 + 100 − 144 = 37
Bottom: 2 × 9 × 10 = 180 - Divide to find cos A. This gives the cosine of the angle as a decimal.cos A = 37180 = 0.20555…
- Use the inverse cosine. The inverse button (cos−1) turns the cosine value back into an angle. (A negative cosine gives an obtuse angle, which is correct.)A = cos−1(0.20555…) = 78.13797…°
- Round to 1 d.p. Give the angle to the accuracy asked.A = 78.1°
Check. Put the angle back in the cosine rule: 92 + 102 − 2 × 9 × 10 × cos 78.1° = 144, and 122 = 144. ✓
Watch out. In the top line the side a opposite the wanted angle is the one that is subtracted. Work out the top and the bottom separately, and do not divide until both are finished.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. When all three sides are known, rearrange the cosine rule to cos A = (b2 + c2 − a2) ÷ 2bc, then use the inverse cosine.
- Spot the angle we want and the side opposite it. With three sides known we can find any angle. Call the side opposite the wanted angle a.Wanted: angle A, opposite the side a = 6.5 cm.
The other two sides are b = 7.2 cm and c = 4.1 cm. - Write the cosine rule and rearrange it for cos A. We want cos A on its own, so add 2bc cos A to both sides, subtract a2 from both sides, then divide by 2bc.a2 = b2 + c2 − 2bc cos A
2bc cos A = b2 + c2 − a2
cos A = b2 + c2 − a22bc - Put the numbers in the top and the bottom. Work the top and the bottom out separately so you do not make a slip.Top: 7.22 + 4.12 − 6.52 = 51.84 + 16.81 − 42.25 = 26.4
Bottom: 2 × 7.2 × 4.1 = 59.04 - Divide to find cos A. This gives the cosine of the angle as a decimal.cos A = 26.40000000000000659.04 = 0.44715…
- Use the inverse cosine. The inverse button (cos−1) turns the cosine value back into an angle. (A negative cosine gives an obtuse angle, which is correct.)A = cos−1(0.44715…) = 63.43873…°
- Round to 1 d.p. Give the angle to the accuracy asked.A = 63.4°
Check. Put the angle back in the cosine rule: 7.22 + 4.12 − 2 × 7.2 × 4.1 × cos 63.4° = 42, and 6.52 = 42. ✓
Watch out. In the top line the side a opposite the wanted angle is the one that is subtracted. Work out the top and the bottom separately, and do not divide until both are finished.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. When all three sides are known, rearrange the cosine rule to cos A = (b2 + c2 − a2) ÷ 2bc, then use the inverse cosine.
- Spot the angle we want and the side opposite it. With three sides known we can find any angle. Call the side opposite the wanted angle a.Wanted: angle A, opposite the side a = 15 cm.
The other two sides are b = 8 cm and c = 10 cm. - Write the cosine rule and rearrange it for cos A. We want cos A on its own, so add 2bc cos A to both sides, subtract a2 from both sides, then divide by 2bc.a2 = b2 + c2 − 2bc cos A
2bc cos A = b2 + c2 − a2
cos A = b2 + c2 − a22bc - Put the numbers in the top and the bottom. Work the top and the bottom out separately so you do not make a slip.Top: 82 + 102 − 152 = 64 + 100 − 225 = −61
Bottom: 2 × 8 × 10 = 160 - Divide to find cos A. This gives the cosine of the angle as a decimal.cos A = −61160 = −0.38125
- Use the inverse cosine. The inverse button (cos−1) turns the cosine value back into an angle. (A negative cosine gives an obtuse angle, which is correct.)A = cos−1(−0.38125) = 112.41113…°
- Round to 1 d.p. Give the angle to the accuracy asked.A = 112.4°
Check. Put the angle back in the cosine rule: 82 + 102 − 2 × 8 × 10 × cos 112.4° = 225.0, and 152 = 225.0. ✓
Watch out. In the top line the side a opposite the wanted angle is the one that is subtracted. Work out the top and the bottom separately, and do not divide until both are finished.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Check whether you have a matching pair (a side and its opposite angle). If you do, use the sine rule. If you have SAS or SSS, use the cosine rule.
- Say what is given. Two sides and the angle between them are known (SAS). You want the third side.Look at the diagram: the marked values show what is known.
- Look for a matching pair. There is no complete matching pair of a side and its opposite angle, but we have SAS or SSS.No matching pair.
- Choose the rule. Use the cosine rule.
Check. Try the other rule: it would leave two unknowns in one equation. ✓
Watch out. The cosine rule is the one to try when the sine rule cannot start.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Check whether you have a matching pair (a side and its opposite angle). If you do, use the sine rule. If you have SAS or SSS, use the cosine rule.
- Say what is given. Two angles and a side are known. You want another side.Look at the diagram: the marked values show what is known.
- Look for a matching pair. There is a complete matching pair (a side and the angle opposite it).Matching pair found.
- Choose the rule. Use the sine rule.
Check. Try the other rule: it would leave two unknowns in one equation. ✓
Watch out. The cosine rule is the one to try when the sine rule cannot start.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Check whether you have a matching pair (a side and its opposite angle). If you do, use the sine rule. If you have SAS or SSS, use the cosine rule.
- Say what is given. All three sides are known. You want an angle.Look at the diagram: the marked values show what is known.
- Look for a matching pair. There is no complete matching pair of a side and its opposite angle, but we have SAS or SSS.No matching pair.
- Choose the rule. Use the cosine rule.
Check. Try the other rule: it would leave two unknowns in one equation. ✓
Watch out. The cosine rule is the one to try when the sine rule cannot start.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Check whether you have a matching pair (a side and its opposite angle). If you do, use the sine rule. If you have SAS or SSS, use the cosine rule.
- Say what is given. Two sides and an angle opposite one of them are known. You want another angle.Look at the diagram: the marked values show what is known.
- Look for a matching pair. There is a complete matching pair (a side and the angle opposite it).Matching pair found.
- Choose the rule. Use the sine rule.
Check. Try the other rule: it would leave two unknowns in one equation. ✓
Watch out. The cosine rule is the one to try when the sine rule cannot start.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. The cosine rule finds the third side of a triangle when two sides and the angle between them are known: a2 = b2 + c2 − 2bc cos A.
- Find the angle at B. Draw a north line at B. The bearing of the second leg is 100°. The back bearing from B to A is 040° + 180° = 220°. The angle ABC is between the direction BA (220°) and BC (100°).Angle ABC = 220° − 100° = 120°
- Name the sides for the cosine rule. The cosine rule uses a for the side opposite the angle. Here the angle is at B, so we write a = AC, and the two sides next to B are b = AB = 5 km and c = BC = 7 km.a = AC (wanted), b = 5, c = 7, angle = 120°
- Spot the angle and the two sides around it. The cosine rule is used when we know two sides and the angle between them. The side we want is the one opposite that angle.Known: b = 5 km, c = 7 km, angle B = 120° between them.
Wanted: a, opposite angle B. - Write the cosine rule and put the numbers in. Square the wanted side on the left; square the other two sides on the right, then take away 2 × side × side × cos(angle).a2 = b2 + c2 − 2bc cos B
a2 = 52 + 72 − 2 × 5 × 7 × cos 120° - Work out the two squares and add. This is the first part, b2 + c2.52 + 72 = 25 + 49 = 74
- Work out the last part on its own. Multiply everything together first. Do this before subtracting (BIDMAS: multiplication comes before subtraction).cos 120° = −0.5
2 × 5 × 7 × −0.5 = −35 - Subtract. This gives a2.a2 = 74 − −35 = 109
- Square root, then round to 1 d.p.. We want a, not a2.a = √109 = 10.44030…
a = 10.4 km
Check. One side of a triangle is between the difference and the sum of the other two: 2 is less than 10.4, which is less than 12. Angle B is more than 90°, so a2 should be more than b2 + c2: 109 is more than 74. ✓
Watch out. Do 2bc cos A first and then subtract it. A very common mistake is to work out (b2 + c2 − 2bc) first. The rule has a minus sign, even for an obtuse angle (cos of an obtuse angle is negative).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. When all three sides are known, rearrange the cosine rule to cos A = (b2 + c2 − a2) ÷ 2bc, then use the inverse cosine.
- Draw the triangle. The footballer at P and the two goalposts G1 and G2 make a triangle. The angle at P is the angle the goal makes at P. The three sides are all known: PG1 = 18 m, PG2 = 20 m and the goal width G1G2 = 7.32 m.Wanted angle: the angle at P, opposite the goal width.
Names for the rule: a = 7.32 (opposite P), b = 18, c = 20. - Spot the angle we want and the side opposite it. With three sides known we can find any angle. Call the side opposite the wanted angle a.Wanted: angle P, opposite the side a = 7.32 m.
The other two sides are b = 18 m and c = 20 m. - Write the cosine rule and rearrange it for cos P. We want cos P on its own, so add 2bc cos P to both sides, subtract a2 from both sides, then divide by 2bc.a2 = b2 + c2 − 2bc cos P
2bc cos P = b2 + c2 − a2
cos P = b2 + c2 − a22bc - Put the numbers in the top and the bottom. Work the top and the bottom out separately so you do not make a slip.Top: 182 + 202 − 7.322 = 324 + 400 − 53.5824 = 670.4176
Bottom: 2 × 18 × 20 = 720 - Divide to find cos P. This gives the cosine of the angle as a decimal.cos P = 670.4176720 = 0.93113…
- Use the inverse cosine. The inverse button (cos−1) turns the cosine value back into an angle. (A negative cosine gives an obtuse angle, which is correct.)P = cos−1(0.93113…) = 21.38747…°
- Round to 1 d.p. Give the angle to the accuracy asked.P = 21.4°
Check. Put the angle back in the cosine rule: 182 + 202 − 2 × 18 × 20 × cos 21.4° = 53.6, and 7.322 = 53.6. ✓
Watch out. In the top line the side a opposite the wanted angle is the one that is subtracted. Work out the top and the bottom separately, and do not divide until both are finished.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. The cosine rule finds the third side of a triangle when two sides and the angle between them are known: a2 = b2 + c2 − 2bc cos A.
- Pick the triangle. The diagonal BD joins B and D. In triangle ABD we know AB = 9, AD = 6 and the angle A = 70° between them (SAS), so the cosine rule gives BD. The other diagonal AC would be longer because it is opposite the 110° angle.Triangle ABD: sides 9 and 6, included angle 70°.
Names for the rule: a = BD (wanted), b = AD = 6, c = AB = 9. - Spot the angle and the two sides around it. The cosine rule is used when we know two sides and the angle between them. The side we want is the one opposite that angle.Known: b = 6 cm, c = 9 cm, angle A = 70° between them.
Wanted: a, opposite angle A. - Write the cosine rule and put the numbers in. Square the wanted side on the left; square the other two sides on the right, then take away 2 × side × side × cos(angle).a2 = b2 + c2 − 2bc cos A
a2 = 62 + 92 − 2 × 6 × 9 × cos 70° - Work out the two squares and add. This is the first part, b2 + c2.62 + 92 = 36 + 81 = 117
- Work out the last part on its own. Multiply everything together first. Do this before subtracting (BIDMAS: multiplication comes before subtraction).cos 70° = 0.34202…
2 × 6 × 9 × 0.34202… = 36.93817… - Subtract. This gives a2.a2 = 117 − 36.93817… = 80.06182…
- Square root, then round to 1 d.p.. We want a, not a2.a = √80.06182… = 8.94772…
a = 8.9 cm
Check. One side of a triangle is between the difference and the sum of the other two: 3 is less than 8.9, which is less than 15. Angle A is less than 90°, so a2 should be less than b2 + c2: 80.06182… is less than 117. ✓
Watch out. Do 2bc cos A first and then subtract it. A very common mistake is to work out (b2 + c2 − 2bc) first. The rule has a minus sign, even for an obtuse angle (cos of an obtuse angle is negative).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Put angle A = 90° into the cosine rule and use cos 90° = 0.
- Write the cosine rule. The side a is opposite angle A.a2 = b2 + c2 − 2bc cos A
- Put in A = 90°. The exact value cos 90° = 0.a2 = b2 + c2 − 2bc × 0
- Simplify. Anything multiplied by 0 is 0.a2 = b2 + c2
- Say what it shows. This is Pythagoras’ theorem, where a is the hypotenuse.The cosine rule includes Pythagoras’ theorem as the special case A = 90°.
Check. Test with b = 3, c = 4, A = 90°: a2 = 9 + 16 = 25, so a = 5. ✓
Watch out. The cosine rule works for every triangle. Pythagoras is the right-angled one.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Compare Zoe’s line with the cosine rule, which has a minus sign in front of the whole of 2bc cos A.
- Find Zoe’s mistake. Zoe worked out (b2 + c2 − 2bc) first, and then multiplied by cos 50°. That is not what the rule says.Zoe: (49 + 81 − 126) × cos 50° = 4 × 0.64278… = 2.57115…
- Work out the correct value. Multiply 2 × 7 × 9 × cos 50° first, then subtract from 49 + 81.2 × 7 × 9 × cos 50° = 126 × 0.64278… = 80.99123…
a2 = 130 − 80.99123… = 49.00876… - Find a. Square root, then round to 1 decimal place.a = √49.00876… = 7.00062…
a = 7.0 cm
Check. The value of a2 should be between (9 − 7)2 = 4 and (9 + 7)2 = 256: 49.00876… fits. Zoe’s 2.57115… is too small to be a2. ✓
Watch out. Always multiply 2bc cos A to one number before you subtract.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. When all three sides are known, rearrange the cosine rule to cos A = (b2 + c2 − a2) ÷ 2bc, then use the inverse cosine.
- Find the largest angle. The largest angle is opposite the longest side, which is 14 m. Call that angle A, so a = 14 and the other sides are b = 11 and c = 8.Wanted: the angle opposite 14 m.
- Spot the angle we want and the side opposite it. With three sides known we can find any angle. Call the side opposite the wanted angle a.Wanted: angle A, opposite the side a = 14 m.
The other two sides are b = 11 m and c = 8 m. - Write the cosine rule and rearrange it for cos A. We want cos A on its own, so add 2bc cos A to both sides, subtract a2 from both sides, then divide by 2bc.a2 = b2 + c2 − 2bc cos A
2bc cos A = b2 + c2 − a2
cos A = b2 + c2 − a22bc - Put the numbers in the top and the bottom. Work the top and the bottom out separately so you do not make a slip.Top: 112 + 82 − 142 = 121 + 64 − 196 = −11
Bottom: 2 × 11 × 8 = 176 - Divide to find cos A. This gives the cosine of the angle as a decimal.cos A = −11176 = −0.0625
- Use the inverse cosine. The inverse button (cos−1) turns the cosine value back into an angle. (A negative cosine gives an obtuse angle, which is correct.)A = cos−1(−0.0625) = 93.58332…°
- Round to 1 d.p. Give the angle to the accuracy asked.A = 93.6°
Check. Put the angle back in the cosine rule: 112 + 82 − 2 × 11 × 8 × cos 93.6° = 196, and 142 = 196. ✓
Watch out. In the top line the side a opposite the wanted angle is the one that is subtracted. Work out the top and the bottom separately, and do not divide until both are finished.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Use the cosine rule for the angle A with the three sides given. If the answer is exactly 0.5 then A = 60°.
- Write cos A. The side opposite A is BC = 7. The other two sides are AB = 5 and AC = 8.cos A = 52 + 82 − 722 × 5 × 8
- Work out the top and bottom. 25 + 64 − 49 on the top, and 2 × 5 × 8 on the bottom.cos A = 25 + 64 − 4980 = 4080 = 12
- Use the exact value. We know cos 60° = ½.So A = 60°.
Check. Calculator: cos−1(0.5) = 60°. ✓
Watch out. In a “show that” question the answer is given: show the working and state the conclusion.