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1.5 Trigonometry

1.5 Trigonometry

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Question 136

The power output PPP, in kilowatts, of a rotational component is modeled by the function

P(θ)=14sin⁡θcos⁡θ+6cos⁡2θ−5 P(\theta) = 14 \sin \theta \cos \theta + 6 \cos^2 \theta - 5 P(θ)=14sinθcosθ+6cos2θ−5

where θ \theta\,θ is the angular displacement in radians for 0≤θ<2π0 \le \theta < 2\pi0≤θ<2π.

a.

Write P(θ)P(\theta)P(θ) in the form asin⁡2θ+bcos⁡2θ+ca \sin 2\theta + b \cos 2\theta + casin2θ+bcos2θ+c, where a,b, a, b,\,a,b, and c c\,c are integers to be found.

[3]
b.

Using your result from part (a), express P(θ)P(\theta)P(θ) in the form Rsin⁡(2θ+α)+cR \sin (2\theta + \alpha) + cRsin(2θ+α)+c where R>0 R > 0\,R>0 and 0<α<π2\displaystyle 0 < \alpha < \frac{\pi}{2}0<α<2π​. Give the exact value of R R\,R and the value of α \alpha\,α in radians to 3 significant figures.

[3]
c.

Hence, or otherwise, (i) state the maximum value of P(θ)P(\theta)P(θ) predicted by this model, (ii) find the second smallest positive value of θ \theta\,θ at which this maximum power output occurs. Give your answer to 3 significant figures.

[4]
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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