Given: $\sin \theta=-\frac{1}{4}$ and $\cos \theta>0$ . Find $\tan \theta$

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Algebra 2

Evaluate expression when two complex numbers are given

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Q. Given:

\sin \theta=-\frac{1}{4}

and

\cos \theta>0

. Find

\tan \theta

Identify Trigonometric Function: We know that $\tan \theta = \frac{\sin \theta}{\cos \theta}$ . We have $\sin \theta = -\frac{1}{4}$ .
Apply Pythagorean Identity: To find $\cos \theta$ , we use the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$ .
Substitute $\sin \theta$ : Substitute $\sin \theta = -\frac{1}{4}$ into the identity: $(-\frac{1}{4})^2 + \cos^2 \theta = 1$ .
Calculate squared value: Calculate $(-\frac{1}{4})^2$ : $(\frac{1}{16}) + \cos^2 \theta = 1$ .
Rearrange for cos theta: Rearrange to find $\cos^2 \theta$ : $\cos^2 \theta = 1 - \frac{1}{16}$ .
Find positive root: Simplify $1 - \frac{1}{16}$ : $\cos^2 \theta = \frac{15}{16}$ .
Divide $\sin$ by $\cos$ : Take the square root of both sides to find $\cos \theta$ . Since $\cos \theta > 0$ , we choose the positive root: $\cos \theta = \sqrt{\frac{15}{16}}$ .
Simplify the division: Simplify $\sqrt{\frac{15}{16}}$ : $\cos \theta = \frac{\sqrt{15}}{4}$ .
Rationalize the denominator: Now, divide $\sin \theta$ by $\cos \theta$ to find $\tan \theta$ : $\tan \theta = \frac{-1}{4} / \frac{\sqrt{15}}{4}$ .
Finalize the solution: Simplify the division: $\tan \theta = -\frac{1}{\sqrt{15}}$ .
Finalize the solution: Simplify the division: $\tan \theta = -\frac{1}{\sqrt{15}}$ .Rationalize the denominator: $\tan \theta = -\frac{1}{\sqrt{15}} \times \frac{\sqrt{15}}{\sqrt{15}}$ .
Finalize the solution: Simplify the division: $\tan \theta = -\frac{1}{\sqrt{15}}$ .Rationalize the denominator: $\tan \theta = -\frac{1}{\sqrt{15}} \times \frac{\sqrt{15}}{\sqrt{15}}$ .Multiply numerator and denominator by $\sqrt{15}$ : $\tan \theta = -\frac{\sqrt{15}}{15}$ .