First slide

Trigonometric Identities

Question

If $\sin θ + \cos θ = p$ and $\sin^{3} θ + \cos^{3} θ = q$ , then $p (p^{2} - 3) =$

Moderate

A

q
B

2q
C

-q
D

-2q

Solution

$\cos θ + \sin θ = p \Rightarrow p^{2}$ $= 1 + 2 \sin θ \cos θ \Rightarrow \sin θ \cos θ = \frac{p^{2} - 1}{2}$ also
$p^{3} = \cos^{3} θ + \sin^{3} θ + 3 \cos θ \sin θ (\cos θ + \sin θ) \Rightarrow p^{3} = q + 3 (\frac{p^{2} - 1}{2}) p$
$\Rightarrow 2 p^{3} = 2 q + 3 p^{3} - 3 p \Rightarrow p^{3} - 3 p = - 2 q$

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