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1.7 Trigonometry

1.7 Trigonometry

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Question 162

In the analysis of a resonant electronic circuit, the phase shift α \alpha\,α across a specific component is related to the impedance through a series of trigonometric relationships.

a.

Prove that

cot⁡α−tan⁡α≡2cot⁡2α \cot \alpha - \tan \alpha \equiv 2 \cot 2\alpha cotα−tanα≡2cot2α

for α≠nπ2,n∈Z\displaystyle \alpha \neq \frac{n\pi}{2}, n \in \mathbb{Z}α=2nπ​,n∈Z.

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b.

Using the identity in part (a), or otherwise, establish that

cot⁡2α−tan⁡2α≡4cot⁡2αcsc⁡2α \cot^2 \alpha - \tan^2 \alpha \equiv 4 \cot 2\alpha \csc 2\alpha cot2α−tan2α≡4cot2αcsc2α
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c.

A particular resonance condition occurs when the operating phase ϕ \phi\,ϕ satisfies the equation

4cot⁡2ϕcsc⁡2ϕ=15tan⁡2ϕ 4 \cot 2\phi \csc 2\phi = 15 \tan^2 \phi 4cot2ϕcsc2ϕ=15tan2ϕ

Solve this equation for −π2<ϕ<π2\displaystyle -\frac{\pi}{2} < \phi < \frac{\pi}{2}−2π​<ϕ<2π​, giving your answers to 2 decimal places.

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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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