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The angle $\theta_1$ is located in Quadrant $\text{III}$ , and $\cos(\theta_1)=-\dfrac{5}{8}$ . What is the value of $\sin(\theta_1)$ ? Express your answer exactly. $\sin(\theta_1)=$

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Full solution

Q. The angle

\theta_1

is located in Quadrant

\text{III}

, and

\cos(\theta_1)=-\dfrac{5}{8}

. What is the value of

\sin(\theta_1)

? Express your answer exactly.

\sin(\theta_1)=

Identify Quadrant and Cosine: Identify the quadrant and cosine value. $\newline$ $\theta_1$ is in Quadrant III, so $\cos(\theta_1) = -\dfrac{5}{8}$ .
Use Pythagorean Identity: Use the Pythagorean identity: $\sin^2(\theta_1) + \cos^2(\theta_1) = 1$ . $\newline$ $\cos^2(\theta_1) = \left(-\dfrac{5}{8}\right)^2 = \dfrac{25}{64}$ .
Calculate Sin Squared: Calculate $\sin^2(\theta_1)$ . $\newline$ $\sin^2(\theta_1) = 1 - \cos^2(\theta_1) = 1 - \dfrac{25}{64} = \dfrac{64}{64} - \dfrac{25}{64} = \dfrac{39}{64}$ .
Find Sin: Find $\sin(\theta_1)$ . $\newline$ $\sin(\theta_1) = \pm\sqrt{\dfrac{39}{64}} = \pm\dfrac{\sqrt{39}}{8}$ .
Determine Sign: Determine the sign of $\sin(\theta_1)$ in Quadrant III. $\newline$ In Quadrant III, $\sin(\theta_1)$ is negative, so $\sin(\theta_1) = -\dfrac{\sqrt{39}}{8}$ .

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