If $\sin \theta =\frac{12}{13}$ and $\theta$ is an acute angle, what is $\cos \theta$?

Step-by-step solution

1. Use the Pythagorean identity ${\sin }^2\theta +{\cos }^2\theta =1$.
2. Square the given sine: ${\sin }^2\theta ={\left(\frac{12}{13}\right)}^2=\frac{144}{169}$.
3. Substitute into the identity: $\frac{144}{169}+{\cos }^2\theta =1$.
4. Subtract $\frac{144}{169}$ from both sides: ${\cos }^2\theta =1-\frac{144}{169}=\frac{25}{169}$.
5. Take the positive square root because $\theta$ is acute: $\cos \theta =\frac{5}{13}$.

Concept tested: Pythagorean identity

Published 25 August 2026 · How this solution was prepared and checked

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