A trig equation (quadratic in sin)

Trigonometry

A-level Maths (9MA0)

ORIGINAL

Spot the hidden quadratic in sin x, solve it, then find every angle in the range.

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Question

Solve

2\sin^2 x + \sin x - 1 = 0

for

0 \le x < 360^\circ

Every step worked, with the reasoning.

1
Let $s = \sin x$ : $2s^2 + s - 1 = 0$ .
It's a quadratic in $\sin x$ .
2
$(2s - 1)(s + 1) = 0$
Factorise the quadratic.
3
$\sin x = \dfrac{1}{2}$ or $\sin x = -1$
Solve each factor for $\sin x$ .
4
$\sin x = \tfrac{1}{2}:\ \ x = 30^\circ,\ 150^\circ$
Principal value $30^\circ$ , and $180^\circ - 30^\circ = 150^\circ$ , both in range.
5
$\sin x = -1:\ \ x = 270^\circ$
The only angle in range with $\sin x = -1$ .

Answer: $x = 30^\circ,\ 150^\circ,\ 270^\circ$