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1.7 Trigonometry

1.7 Trigonometry

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Question 145

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

1.7 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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