First slide

Trigonometric Identities

Question

If $x = a (\cos θ + θ \sin θ), y = a (\sin θ - θ \cos θ)$ then $a θ =$

Moderate

A

$x + y - a$
B

$\sqrt{x^{2} + y^{2} - a^{2}}$
C

$\sqrt{x^{2} - y^{2} + a^{2}}$
D

$x - y + a$

Solution

$\begin{array}{l} x^{2} + y^{2} = a^{2} [\cos^{2} θ + \sin^{2} θ + θ^{2} \sin^{2} θ + θ^{2} \cos^{2} θ] \\ = a^{2} (1 + θ^{2}) \\ \Rightarrow a θ = \sqrt{x^{2} + y^{2} - a^{2}} \end{array}$

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