First slide

Trigonometric Identities

Question

If $\sin A = \sin^{2} B and 2 \cos^{2} A = 3 \cos^{2} B$ then

Moderate

A

$\sin A = \frac{1}{2}$
B

$\cos B = \frac{1}{\sqrt{2}}$
C

$\sin C = \frac{\sqrt{3}}{2}$
D

$s i n C = \frac{\sqrt{3} + 1}{2}$

Solution

$\begin{array}{l} \sin A = \sin^{2} B and 2 \cos^{2} A = 3 \cos^{2} B \\ \Rightarrow 2 - 2 \sin^{2} A = 3 - 3 \sin^{2} B \\ \Rightarrow 2 \sin^{2} A - 3 \sin A + 1 = 0 \\ \Rightarrow (2 \sin A - 1) (\sin A - 1) = 0 \\ \Rightarrow A = 30^{\circ} or A = 90^{\circ} \end{array}$
if $A = 30^{\circ} \Rightarrow B = 45^{\circ} \Rightarrow C = 105^{\circ}$
if $A - 90^{n} \Rightarrow B = 90^{\circ}$ which is not possible

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