1.8 E: Trigonometry

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Question 71

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

The configuration of the arm is defined by an angle $\phi\,$ in the range $\le \phi \le \pi$ . The coordinates of $P\,$ are $\cos \phi, 2k \sin \phi)$ and the position of $Q\,$ relative to $P\,$ is $\cos 2\phi, k \sin 2\phi)$ .

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

k(2\cos \phi + \cos 2\phi)

[2]

Using a double angle identity, show that

k(2\cos^2 \phi + 2\cos \phi - 1)

[2]

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

[3]

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

[3]

1.8 E: Trigonometry

Question 71

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

k(2\cos \phi + \cos 2\phi)

[2]

Using a double angle identity, show that

k(2\cos^2 \phi + 2\cos \phi - 1)

[2]

[3]

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

[3]

1.8 E: Trigonometry

Question 71

Markscheme

1.8 E: Trigonometry

Question 71

Markscheme

1.8 E: Trigonometry

Question 71

1.8 E: Trigonometry Questions

1.8 E: Trigonometry

Question 71

1.8 E: Trigonometry Questions