The sine of an obtuse angle is $5/13$. Without the use of a calculator, find the value for the cosine of the same angle. Answer ...

1. Find the magnitude of the cosine using the Pythagorean identity.

Let the obtuse angle be $\theta$.

$\sin^2\theta+\cos^2\theta=1$

Given $\sin\theta=\dfrac{5}{13}$,

$\left(\dfrac{5}{13}\right)^2+\cos^2\theta=1$

$\cos^2\theta=1-\dfrac{25}{169}=\dfrac{144}{169}$

$|\cos\theta|=\dfrac{12}{13}$

2. Choose the correct sign for cosine of an obtuse angle.

Since $\theta$ is obtuse, $90^\circ<\theta<180^\circ$.

In this range, cosine is negative.

$\cos\theta=-\dfrac{12}{13}$