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1.5 Trigonometry

1.5 Trigonometry

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Question 148

A robotic tracking arm consists of two segments, OA OA\,OA and ABABAB, each of length 16 cm. The segment OA OA\,OA is pivoted at the origin OOO. The configuration of the arm is defined by the angle θ \theta\,θ that OA OA\,OA makes with the positive xxx-axis.

The arm is geared such that the second segment AB AB\,AB always makes an angle of 2θ 2\theta\,2θ with the positive xxx-axis, where 0≤θ≤π2\displaystyle 0 \le \theta \le \frac{\pi}{2}0≤θ≤2π​.

a.

Show that the xxx-coordinate of the end effector B B\,B is given by

x=16(cos⁡θ+cos⁡2θ) x = 16(\cos \theta + \cos 2\theta) x=16(cosθ+cos2θ)
[3]
b.

Using a double angle identity, show that

x=16(2cos⁡2θ+cos⁡θ−1) x = 16(2\cos^2 \theta + \cos \theta - 1) x=16(2cos2θ+cosθ−1)
[2]
c.

The expression in part (b) can be written in the form x=32(cos⁡θ+14)2−18\displaystyle x = 32\left(\cos \theta + \frac{1}{4}\right)^2 - 18x=32(cosθ+41​)2−18. Determine the value of θ \theta\,θ for which x x\,x is a minimum, and calculate this minimum value.

[3]
d.

Calculate the exact distance OB OB\,OB from the origin to the end effector when x x\,x is at its minimum value.

[2]
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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