# Pythagorean identity review

CCSS Math: HSF.TF.C.8
Review the Pythagorean trigonometric identity and use it to solve problems.

## What is the Pythagorean identity?

$\sin^2(\theta)+\cos^2(\theta)=1$
This identity is true for all real values of $\theta$. It is a result of applying the Pythagorean theorem on the right triangle that is formed in the unit circle for each $\theta$.

## What problems can I solve with the Pythagorean identity?

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 $\theta$ in Quadrant $\text{IV}$ for which $\sin(\theta)=-\dfrac{24}{25}$. We can use the Pythagorean identity and $\sin(\theta)$ to solve for $\cos(\theta)$:
\begin{aligned} \sin^2(\theta)+\cos^2(\theta)&=1 \\\\ \left(-\dfrac{24}{25}\right)^2+\cos^2(\theta)&=1 \\\\ \cos^2(\theta)&=1-\left(-\dfrac{24}{25}\right)^2 \\\\ \sqrt{\cos^2(\theta)}&=\sqrt\dfrac{49}{625} \\\\ \cos(\theta)&=\pm\dfrac{7}{25} \end{aligned}
The sign of $\cos(\theta)$ is determined by the quadrant. $\theta$ is in Quadrant $\text{IV}$, so its cosine value must be positive. In conclusion, $\cos(\theta)=\dfrac{7}{25}$.
Problem 1
$\theta_1$ is located in Quadrant $\text{III}$, and $\cos(\theta_1)=-\dfrac{3}{5}$ .
$\sin(\theta_1)=$