“Teach A Level Maths” Vol. 1: AS Core Modules 36: The Cosine Rule

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The cosine rule is used to find sides and angles of a scalene triangle when 2 sides and the angle formed by them are known, or all 3 sides are known In both these cases, we don’t know a pair of side and opposite angle so the sine rule cannot be used. We will now prove the cosine rule but you do not need to learn the proof.

c h x c - x Proof of the Cosine Rule In the triangle ABC, draw the perpendicular, h, from C to AB. A B C b a c Let AN = x. Then, NB = c - x. From triangle ANC, N h x c - x Using Pythagoras’ theorem: h In triangle ANC, h In triangle BNC, So,

Proof of the Cosine Rule We have Simplifying: Substituting for x from equation (1), Rearranging:

The Cosine Rule for triangle ABC We use this arrangement when 2 sides and the angle formed by them are known. The letters can be switched to find any side provided it is opposite the given angle.

The Cosine Rule for triangle ABC We use this arrangement when 2 sides and the angle formed by them are known. The letters can be switched to find any side provided it is opposite the given angle. If we want to find an angle, we use the sine rule after we have used the cosine rule.

b c a 19 e.g. Find side c and angle B in the triangle ABC A B C 15 Solution: Use the Cosine rule ( 3 s.f.) Tip: Do the whole calculation in one go on your calculator. It avoids errors! The Sine rule: Tip: Leave the answer on your calculator as it will be needed to find angle B ( 3 s.f.)

Exercise 7 1. Find p in the triangle PQR P R Q 6 p Solution: ( 3 s.f.)

The 2nd form of the Cosine Rule We know that Rearranging, The minus sign . . . . . . belongs to the side opposite the angle we are finding We use this form to find any angle of a triangle when we know all 3 sides.

The Cosine Rule 6 e.g. 1 Find angle X in triangle XYZ Y Z 8 4 X Solution: Use the Cosine Rule

C 9 A e.g. 2 Find all the angles in triangle ABC 6 5 B Solution: Let’s find A first We can now use the Cosine rule again or switch to the Sine rule. If we use the Sine rule, we must avoid the largest angle ( opposite the longest side ) as we don’t know whether it is less than or greater than .

5 C 9 A 6 B EITHER: Using the Cosine rule for B or C: e.g. OR: Using the Sine rule for C :

SUMMARY The Cosine Rule In a triangle that isn’t right angled, if we know 2 sides and the angle formed by the 2 sides, we use to find the 3rd side. If we know 3 sides, we use to find any angle.

Exercise 7 1. Find all the angles in the triangle XYZ giving the answers to 1 decimal place. Y Z X 4 9 Solution: Use the Cosine rule for any angle. e.g. (1 d. p.)

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The cosine rule is used to find sides and angles of a scalene triangle when 2 sides and the angle formed by them are known, or all 3 sides are known In both these cases, we don’t know a side and its opposite angle so the sine rule cannot be used.

The letters can be switched to find any side provided it is opposite the given angle. The Cosine Rule for triangle ABC We use this arrangement when 2 sides and the angle formed by them are known. If we want to find an angle, we use the sine rule after we have used the cosine rule.

b c a A e.g. Find side c and angle B in the triangle ABC 15 19 e.g. Find side c and angle B in the triangle ABC A B C 15 c ( 3 s.f.) Solution: Use the Cosine rule The Sine rule: Tip: Do the whole calculation in one go on your calculator and leave your answer so it can be used to find B. a b

The 2nd form of the Cosine Rule We know that Rearranging, We use this form to find any angle of a triangle when we know all 3 sides. The minus sign goes with the side opposite the angle we are finding.

Solution: Let’s find A first 6 e.g. 2 Find all the angles in triangle ABC B A 9 5 C We can now use the Cosine rule again or switch to the Sine rule. If we use the Sine rule, we must avoid the largest angle ( opposite the longest side ) as we don’t know whether it is less than or greater than .

OR: Using the Sine rule for C : EITHER: Using the Cosine rule for B or C: e.g. 6 B A 9 5 C

The Cosine Rule SUMMARY In a triangle that isn’t right angled, if we know 2 sides and the angle formed by the 2 sides, we use If we know 3 sides, we use to find the 3rd side. to find any angle.