Trigonometry (1) Sine Rule Cosine Rule

[Height] h2 = b2 - x2 [Height] h2 = a2 - (c-x)2 h2 = a2 - c2 +2cx - x2 C b2 - x2 = a2 - c2 +2cx - x2 +x2 +x2 a b2 = a2 - c2 +2cx b h b2 = a2 - c2 +2c x b cos A a2 = b2 + c2 - 2bc cos A B A x c-x c cos A = Adj/Hyp = x/b x = b cos A The cosine rule proof - won’t be examined

The Cosine Rule a2 = b2 + c2 - 2bc cos A b2 = a2 + c2 - 2ac cos B … is used for working out angles and sides in non-right angled triangles a2 = b2 + c2 - 2bc cos A It is …. C angles a sides b A B c By similar proofs:- b2 = a2 + c2 - 2ac cos B c2 = a2 + b2 - 2ab cos C

The Cosine Rule - example Finding a side a2 = b2 + c2 - 2bc cos A C angles sides 6 a ? 75o A B 8 a2 = 62 + 82 - 2x6x8 cos 75 a2 = 36 + 64 - 96 x 0.2588 a2 = 75.153 a =8.67 cm [2 d.p.]

The Cosine Rule - example Finding an angle a2 = b2 + c2 - 2bc cos A C angles sides 7.1 13.5 ? A B 8.8 13.52 = 7.12 + 8.82 - 2x7.1x8.8 cos A 2x7.1x8.8 cos A = 7.12 + 8.82 - 13.52 cos A = 7.12 + 8.82 - 13.52 = -0.4353 2x7.1x8.8 A = 116o [to nearest degree]

Turn to page 71 of your Core 2 book and answer … Activity Turn to page 71 of your Core 2 book and answer … exercise B 1a) 2a)

sin A = Opp/Hyp = h/b C h = b sin A sin B = Opp/Hyp = h/a h = a sin B a If the perpendicular was here b h h = b sin A = a sin B B sin A sin B a = b A c The sine rule proof - won’t be examined sin B sin C b = c

The Sine Rule It is …. a = b = c sin A sin B sin C C angles a sides b … is used for working out angles and sides in non-right angled triangles It is …. a = b = c sin A sin B sin C C angles a sides b A B c

The Sine Rule - example a = b = c sin A sin B sin C a = b a = 4 Finding a side a = b = c sin A sin B sin C C angles sides 4 cm a ? 75o 35o A B a = b sin A sin B a = 4 sin 35 sin 75 a = 4 x sin 35 = 2.375 = 2.4 cm [1dp] sin 75

The Sine Rule - example a = b = c sin A sin B sin C a = b 4 = 6.5 Finding an angle a = b = c sin A sin B sin C C angles sides 6.5 cm 4 cm 85o ? A B a = b sin A sin B 4 = 6.5 sin A sin 85 4 x sin 85 = 6.5 x sin A sin A = 4 x sin 85 = 0.613 6.5 A = 38o or 180-38 = 162o

Turn to page 74 of your Core 2 book and answer … Activity Turn to page 74 of your Core 2 book and answer … exercise C 1a) 3a)