First slide

Trigonometric equations

Question

$I f α, β a r e s o l u t i o n s o f \sin^{2} x + a \sin x + b = 0 a n d \cos^{2} x + c \cos x + d = 0$ , then $\sin (α + β)$ is equal to

Moderate

A

$\frac{2 a c}{a^{2} + c^{2}}$
B

$\frac{a^{2} + c^{2}}{2 a c}$
C

$\frac{2 b d}{b^{2} + d^{2}}$
D

$\frac{b^{2} + d^{2}}{2 b d}$

Solution

$W e h a v e \sin α + \sin β = - a, \cos α + \cos β = - c \Rightarrow \frac{\sin α + \sin β}{\cos α + \cos β} = \frac{a}{c} \Rightarrow \frac{2 \sin (\frac{α + β}{2}) \cos (\frac{α - β}{2})}{2 \cos (\frac{α + β}{2}) \cos (\frac{α - β}{2})} = \frac{a}{c} \Rightarrow \tan (\frac{α + β}{2}) = \frac{a}{c} N o w \sin (α + β) = \frac{2 \tan (\frac{α + β}{2})}{1 + \tan^{2} (\frac{α + β}{2})} = \frac{2 a c}{a^{2} + c^{2}}$

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