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

The Cosine Rule Module C2 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

The Cosine Rule 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.

The Cosine Rule In the triangle ABC, draw the perpendicular, h, from C to AB. Proof of the Cosine Rule N h Let AN = x. Then, NB = c  x. xc - x Using Pythagoras’ theorem: In triangle ANC, In triangle BNC, So, From triangle ANC, AB C b a c h2h2 h2h2

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

The Cosine Rule 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 Cosine Rule 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.

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

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

The Cosine Rule... belongs to the side opposite the angle we are finding The 2 nd 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...

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

The Cosine Rule 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.

The Cosine Rule 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  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 3 rd side. to find any angle.

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

The Cosine Rule

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

The Cosine Rule 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 Cosine Rule 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.

The Cosine Rule 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: ( 3 s.f.) 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 Cosine Rule The 2 nd 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.

The Cosine Rule 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.

The Cosine Rule 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  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 3 rd side. to find any angle.