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Trigonometry (1)

Trigonometry (1). Sine Rule Cosine Rule. x. c-x. [Height] h 2 = b 2 - x 2. [Height] h 2 = a 2 - (c-x) 2 h 2 = a 2 - c 2 +2cx - x 2. C. b 2 - x 2 = a 2 - c 2 +2cx - x 2. +x 2. +x 2. b 2 = a 2 - c 2 +2cx. a. b. h. b 2 = a 2 - c 2 +2c x b cos A.

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Trigonometry (1)

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  1. Trigonometry (1) Sine Rule Cosine Rule

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

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

  4. The Cosine Rule - example Finding a side a2 = b2 + c2 - 2bc cosA C angles a ? sides 6 75o B A 8 a2 = 62 + 82 - 2x6x8 cos75 a2 = 36 + 64 - 96 x 0.2588 a2 = 75.153 a =8.67 cm [2 d.p.]

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

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

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

  8. The Sine Rule … 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 B A c

  9. The Sine Rule - example 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 Finding a side a = b = c sin A sin B sin C C angles sides 4 cm a ? 75o 35o B A

  10. The Sine Rule - example 4 = 6.5 sin A sin 85 a = b sin A sin B Finding an angle a = b = c sin A sin B sin C C angles sides 6.5 cm 4 cm 85o ? B A 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

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

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