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1.7 Trigonometry

1.7 Trigonometry

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Question 98
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

1.7 Trigonometry Questions

  1. A Level
  2. /Maths
  3. /1.7 Trigonometry

317 exam-style questions on OCR (MEI) A Level Maths 1.7 Trigonometry, covering 1.7.1 Solve right-angled triangles, 1.7.2 Definitions of sin, cos and tan for any angle, 1.7.3 Graphs of sin, cos and tan, 1.7.4 Exact values of trig functions (degrees), 1.7.5 Area of a triangle, 1.7.6 Sine and cosine rules, 1.7.7 Identity tan = sin/cos, 1.7.8 Identity sin^2 + cos^2 = 1, 1.7.9 Solve simple trigonometric equations, 1.7.10 Exact values of trig functions (radians) (A-level only), 1.7.11 Inverse trigonometric functions (A-level only), 1.7.12 Radians and degree conversion (A-level only), 1.7.13 Arc length and area of a sector (A-level only), 1.7.14 Small angle approximations (A-level only), 1.7.15 Sec, cosec and cot functions (A-level only), 1.7.16 Graphs of reciprocal trig functions (A-level only), 1.7.17 Pythagorean identities for sec and cosec (A-level only), 1.7.18 Compound angle formulae (A-level only), 1.7.19 Double angle identities (A-level only), 1.7.20 Expressions for a cos θ ± b sin θ (A-level only), 1.7.21 Use identities to solve equations (A-level only), 1.7.22 Proofs involving trigonometric functions (A-level only), 1.7.23 Trig to solve problems in context (A-level only), and 1.7 Trigonometry. Each one has a worked solution and a mark scheme showing where the marks go.

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