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If $\cos^2 \theta + \frac{1}{8} = \sin^2 \theta$ , fin...

If $\cos^2 \theta + \frac{1}{8} = \sin^2 \theta$ , find $\tan \theta$ .

A.

3

B.

$\frac{3\sqrt{7}}{7}$

C.

3 $\sqrt{7}$

D.

$\sqrt{7}$

View answer & explanation

Correct answer is B

$\cos^2 \theta + \frac{1}{8} = \sin^2 \theta$ ..........(i)

from trigometric ratios for an acute angle, where $cos^{2} \theta + \sin^2 \theta = 1$ ........(ii)

Substitute for equation (i) in (i) = $\cos^2 \theta + \frac{1}{8} = 1 - \cos^2 \theta$

= $\cos^2 \theta + \cos^2 \theta = 1 - \frac{1}{8}$

$2\cos^2 \theta = \frac{7}{8}$

$\cos^2 \theta = \frac{7}{2 \times 8}$

$\frac{7}{16} = \cos \theta$

$\sqrt{\frac{7}{16}}$ = $\frac{\sqrt{7}}{4}$

but cos $\theta$ = $\frac{\text{adj}}{\text{hyp}}$

$opp^2 = hyp^2 - adj^2$

$opp^2 = 4^2 - (\sqrt{7})^{2}$

= 16 - 7

opp = $\sqrt{9}$ = 3

$\tan \theta = \frac{\text{opp}}{\text{adj}}$

= $\frac{3}{\sqrt{7}}$

$\frac{3}{\sqrt{7}}$ x $\frac{\sqrt{7}}{\sqrt{7}}$ = $\frac{3\sqrt{7}}{7}$

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