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

What is the Pythagorean identity?

sin2(θ)+cos2(θ)=1\sin^2(\theta)+\cos^2(\theta)=1
This identity is true for all real values of θ\theta. It is a result of applying the Pythagorean theorem on the right triangle that is formed in the unit circle for each θ\theta.
Want to learn more about the Pythagorean identity? Check out this video.

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 θ\theta in Quadrant IV\text{IV} for which sin(θ)=2425\sin(\theta)=-\dfrac{24}{25}. We can use the Pythagorean identity and sin(θ)\sin(\theta) to solve for cos(θ)\cos(\theta):
sin2(θ)+cos2(θ)=1(2425)2+cos2(θ)=1cos2(θ)=1(2425)2cos2(θ)=49625cos(θ)=±725\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(θ)\cos(\theta) is determined by the quadrant. θ\theta is in Quadrant IV\text{IV}, so its cosine value must be positive. In conclusion, cos(θ)=725\cos(\theta)=\dfrac{7}{25}.
Problem 1
θ1\theta_1 is located in Quadrant III\text{III}, and cos(θ1)=35\cos(\theta_1)=-\dfrac{3}{5} .
sin(θ1)=\sin(\theta_1)=

Express your answer exactly.

Want to try more problems like this? Check out this exercise.
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