The Cosine Rule

Starter Work out the value of the missing angle, θ. Sinθ 20 Sin 71 22 = θ 22mm Sin 71 22 Sinθ = x 20 71° Sinθ = 0.85956… 20mm Sin-1 θ = 59.3° (1dp)

The Cosine Rule Last lesson we saw the Sine Rule, which linked opposite sides and angles in any triangle This involved the use of Sine This lesson we will learn to use the Cosine Rule, which also allows us to find a missing side or angle…

The Cosine Rule Learning Objectives All will be able to use the cosine rule to calculate missing sides in a triangle (Grade A/A*) Most will also be able to use this rule to calculate missing angles (Grade A*) Some will be able to use this in conjunction with the Sine rule and others in order to answer more problems! (Grade A*)

The Cosine Rule The Cosine Rule states that in any triangle, labelled as to the right; a2 = b2 + c2 - 2bc cosA b2 = a2 + c2 – 2ac cosB c2 = a2 + b2 – 2ab cos C B c a A b C

The Cosine Rule Find the missing side in the triangle opposite. a2 = b2 + c2 – 2bc cosA a2 = 72 + 92 – (2x7x9)cos 81 a2 = 49 + 81 – 126cos81 a2 = 130 – 19.71… a = 10.5cm (1dp) a2 = b2 + c2 - 2bc cosA b a 7cm ? 81º A 9cm c √

The Cosine Rule Find the missing side in the triangle opposite. a2 = b2 + c2 – 2bc cosA a2 = 102 + 182 – (2x10x18)cos 32 a2 = 100 + 324 – 360cos32 a2 = 424 – 305.297… a = 10.9cm (1dp) a2 = b2 + c2 - 2bc cosA A b 10m 32º 18m c ? √ a

The Cosine Rule Find the missing angle in the triangle opposite. a2 = b2 + c2 – 2bc cosA 112 = 132 + 122 – (2x13x12)cosA 121 = 169 + 144 – 312cosA 121 = 313 – 312cosA -192 = - 312cosA 0.615… = cos A A = 52.02º (2dp) a2 = b2 + c2 - 2bc cosA A ? 12m c b 13m - 313 ÷ -312 11m Cos-1 a

The Cosine Rule Find the missing angle in the triangle opposite. a2 = b2 + c2 – 2bc cosA 52 = 112 + 152 – (2x15x11)cosA 25 = 121 + 225 – 330cosA 25 = 346 – 330cosA -321 = - 330cosA 0.972… = cos A A = 13.41º (2dp) a2 = b2 + c2 - 2bc cosA c 15m - 346 a 5m ÷ -330 ? A 11m Cos-1 b

Plenary Find the missing side in the triangle opposite. a2 = b2 + c2 – 2bc cosA a2 = 72 + 142 – (2x7x14)cos90 a2 = 49 + 196 – 196cos90 a2 = 245 – 196cos90 a2 = 245 a = 15.6 (1dp) a2 = b2 + c2 – 2bc cosA (Using the cosine rule) 15.6m a b ? 7m 196Cos90 = 0 A 14m √ c If the angle used is 90, then the part with Cos in will be 0… So a2 = b2 + c2  PYTHAGORAS!

Plenary Learning Objectives All will be able to use the cosine rule to calculate missing sides in a triangle (Grade A/A*) Most will also be able to use this rule to calculate missing angles (Grade A*) Some will be able to use this in conjunction with the Sine rule and others in order to answer more problems! (Grade A*)

Summary We have learnt how to use the Cosine Rule We have seen that it works in any triangle We have used it to find both missing sides and angles