If $\sin \theta =\frac{5}{13}$ and $\theta$ is an acute angle, find $\tan \theta$.

Step-by-step solution

1. Use the identity ${\sin }^2\theta +{\cos }^2\theta =1$ to find $\cos \theta$.
2. $\cos \theta =\sqrt{1-{\left(\frac{5}{13}\right)}^2}=\sqrt{1-\frac{25}{169}}=\sqrt{\frac{144}{169}}=\frac{12}{13}$ because $\theta$ is acute, so cosine is positive.
3. Now $\tan \theta =\frac{\sin \theta }{\cos \theta }=\frac{\frac{5}{13}}{\frac{12}{13}}=\frac{5}{12}$.

Concept tested: Using $\sin^2+\cos^2=1$ to find tangent

Published 22 August 2026 · How this solution was prepared and checked

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