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1.5 Trigonometry

1.5 Trigonometry

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Question 104
a.

Given that cos⁡ϕ≠±1\cos \phi \neq \pm 1cosϕ=±1, prove the identity

11−cos⁡ϕ+11+cos⁡ϕ≡2csc⁡2ϕ \frac{1}{1 - \cos \phi} + \frac{1}{1 + \cos \phi} \equiv 2 \csc^2 \phi 1−cosϕ1​+1+cosϕ1​≡2csc2ϕ
[3]
b.

The response RRR of a specific sensor is modeled by the expression R=11−cos⁡ϕ+11+cos⁡ϕR = \frac{1}{1 - \cos \phi} + \frac{1}{1 + \cos \phi}R=1−cosϕ1​+1+cosϕ1​. Find the set of values of kkk for which the equation R=kR = kR=k has real solutions for ϕ\phiϕ. Fully justify your answer.

[2]
c.

Given that ϕ\phiϕ is in the third quadrant (reflex angle between 180∘180^\circ180∘ and 270∘270^\circ270∘) and

11−cos⁡ϕ+11+cos⁡ϕ=8 \frac{1}{1 - \cos \phi} + \frac{1}{1 + \cos \phi} = 8 1−cosϕ1​+1+cosϕ1​=8

find the exact value of cot⁡ϕ\cot \phicotϕ.

[3]
Markscheme

1.5 Trigonometry Questions

  1. A Level
  2. /Maths
  3. /1.5 Trigonometry

317 exam-style questions on OCR A Level Maths 1.5 Trigonometry, covering 1.5.1 Definitions for all arguments, 1.5.2 Sine and cosine rules, 1.5.3 Area of a triangle, 1.5.4 Radian measure (A-level only), 1.5.5 Small angle approximations (A-level only), 1.5.6 Graphs of basic trigonometric functions, 1.5.7 Exact values in radians (A-level only), 1.5.8 Reciprocal and inverse trigonometric ratios (A-level only), 1.5.9 Graphs of reciprocal and inverse functions (A-level only), 1.5.10 Trigonometric identities, 1.5.11 Further trigonometric identities (A-level only), 1.5.12 Double angle and compound angle formulae (A-level only), 1.5.13 Geometrical proofs of formulae (A-level only), 1.5.14 Harmonic form Rcos / Rsin (A-level only), 1.5.15 Trigonometric equations, 1.5.16 Proof involving trigonometric functions (A-level only), and 1.5.17 Trigonometric functions in context. Each one has a worked solution and a mark scheme showing where the marks go.

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