First slide

Trigonometric equations

Question

General solution of the equation $2 \cos^{2} θ + 11 \sin θ = 7$ is

Moderate

A

$θ = n π + {(- 1)}^{n} \frac{π}{6}$
B

$θ = 2 n π + \frac{π}{6}$
C

$θ = n π + {(- 1)}^{n} \frac{π}{3}$
D

$θ = 2 n π + \frac{π}{3}$

Solution

$\begin{array}{l} W e h a v e 2 \cos^{2} θ + 11 \sin θ = 7 \\ \Rightarrow 2 (1 - \sin^{2} θ) + 11 \sin θ = 7 \\ \Rightarrow 2 \sin^{2} θ - 11 \sin θ + 5 = 0 \\ \Rightarrow (2 \sin θ - 1) (\sin θ - 5) = 0 \\ \Rightarrow \sin θ = \frac{1}{2}, (\sin θ \neq 5) \\ \Rightarrow θ = n π + {(- 1)}^{n} \frac{π}{6}, n \in z \end{array}$

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