Pythagorean identity review

This identity is true for all real values of $\theta$theta. It is a result of applying the Pythagorean theorem on the right triangle that is formed in the unit circle for each $\theta$theta.

Like any identity, the Pythagorean identity can be used for rewriting trigonometric expressions in equivalent, more useful, forms.

The Pythagorean theorem also allows us to convert between the sine and cosine values of an angle, without knowing the angle itself. Consider, for example, the angle $\theta$theta in Quadrant $\text{IV}$I, V for which $\sin(\theta)=-\dfrac{24}{25}$sine, left parenthesis, theta, right parenthesis, equals, minus, start fraction, 24, divided by, 25, end fraction. We can use the Pythagorean identity and $\sin(\theta)$sine, left parenthesis, theta, right parenthesis to solve for $\cos(\theta)$cosine, left parenthesis, theta, right parenthesis:

The sign of $\cos(\theta)$cosine, left parenthesis, theta, right parenthesis is determined by the quadrant. $\theta$theta is in Quadrant $\text{IV}$I, V, so its cosine value must be positive. In conclusion, $\cos(\theta)=\dfrac{7}{25}$cosine, left parenthesis, theta, right parenthesis, equals, start fraction, 7, divided by, 25, end fraction.