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Pythagorean identity review

Review the Pythagorean trigonometric identity and use it to solve problems.

What is the Pythagorean identity?

sine, squared, left parenthesis, theta, right parenthesis, plus, cosine, squared, left parenthesis, theta, right parenthesis, equals, 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.
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 theta in Quadrant start text, I, V, end text for which sine, left parenthesis, theta, right parenthesis, equals, minus, start fraction, 24, divided by, 25, end fraction. We can use the Pythagorean identity and sine, left parenthesis, theta, right parenthesis to solve for cosine, left parenthesis, theta, right parenthesis:
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 cosine, left parenthesis, theta, right parenthesis is determined by the quadrant. theta is in Quadrant start text, I, V, end text, so its cosine value must be positive. In conclusion, cosine, left parenthesis, theta, right parenthesis, equals, start fraction, 7, divided by, 25, end fraction.
Problem 1
theta, start subscript, 1, end subscript is located in Quadrant start text, I, I, I, end text, and cosine, left parenthesis, theta, start subscript, 1, end subscript, right parenthesis, equals, minus, start fraction, 3, divided by, 5, end fraction .
sine, left parenthesis, theta, start subscript, 1, end subscript, right parenthesis, equals

Express your answer exactly.

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

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