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1.7 Trigonometry

1.7 Trigonometry

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Question 206

A robotic surveillance arm is anchored at a fixed pivot O O\,O at the origin of a coordinate system. The arm consists of two segments, OP OP\,OP and PQPQPQ, with lengths 2k 2k\,2k and k k\,k respectively. A sensor is attached at point QQQ.

The configuration of the arm is defined by an angle ϕ \phi\,ϕ in the range 0≤ϕ≤π0 \le \phi \le \pi0≤ϕ≤π. The coordinates of P P\,P are (2kcos⁡ϕ,2ksin⁡ϕ)(2k \cos \phi, 2k \sin \phi)(2kcosϕ,2ksinϕ) and the position of Q Q\,Q relative to P P\,P is (kcos⁡2ϕ,ksin⁡2ϕ)(k \cos 2\phi, k \sin 2\phi)(kcos2ϕ,ksin2ϕ).

a.

Show that the xxx-coordinate of the sensor at Q Q\,Q is given by

x=k(2cos⁡ϕ+cos⁡2ϕ) x = k(2\cos \phi + \cos 2\phi) x=k(2cosϕ+cos2ϕ)
[2]
b.

Using a double angle identity, show that

x=k(2cos⁡2ϕ+2cos⁡ϕ−1) x = k(2\cos^2 \phi + 2\cos \phi - 1) x=k(2cos2ϕ+2cosϕ−1)
[2]
c.

The expression in part (b) can be written as x=2k(cos⁡ϕ+12)2−3k2\displaystyle x = 2k\left(\cos \phi + \frac{1}{2}\right)^2 - \frac{3k}{2}x=2k(cosϕ+21​)2−23k​. Find the value of ϕ \phi\,ϕ for which x x\,x is a minimum and state this minimum value in terms of kkk.

[3]
d.

Calculate the exact distance OQ OQ\,OQ when x x\,x is at its minimum value, giving your answer in terms of kkk.

[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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