The angle theta_(1) is located in Quadrant I, and sin(theta_(1))=(1)/(2). What is the value of cos(theta_(1)) ? Express your answer exactly. cos(theta_(1))=

Use Pythagorean Identity: We know that sin(theta_(1)) = 1/2 and theta_(1) is in Quadrant I. Use the Pythagorean identity sin²(theta) + cos²(theta) = 1 to find cos(theta_(1)). Substitute 1/2 for sin(theta_(1)) in sin²(theta_(1)) + cos²(theta_(1)) = 1. (1/2)² + cos²(theta_(1)) = 1.Substitute and Simplify: Simplify (1/2)² + cos²(theta_(1)) = 1 to find the value of cos(theta_(1)). (1/4) + cos²(theta_(1)) = 1 cos²(theta_(1)) = 1 - 1/4 cos²(theta_(1)) = 3/4Take Square Root: Since we are looking for cos(theta_(1)), we take the square root of both sides. cos(theta_(1)) = ±√(3/4) cos(theta_(1)) = ±√3/2Determine Sign and Final Answer: Determine the sign of cos(theta_(1)) based on the quadrant in which theta_(1) is located. Since theta_(1) is in Quadrant I, where both sine and cosine are positive, we choose the positive value for cos(theta_(1)). cos(theta_(1)) = √3/2

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The angle
theta_(1) is located in Quadrant
I, and
sin(theta_(1))=(1)/(2).
What is the value of
cos(theta_(1)) ?
Express your answer exactly.

cos(theta_(1))=

The angle $\theta_{1}$ is located in Quadrant $\newline$ I, and $\sin \left(\theta_{1}\right)=\frac{1}{2}$ . $\newline$ What is the value of $\cos \left(\theta_{1}\right)$ ? $\newline$ Express your answer exactly. $\newline$ $\cos \left(\theta_{1}\right)=$

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Precalculus

Find trigonometric ratios using multiple identities

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Q. The angle

\theta_{1}

is located in Quadrant

\newline

I, and

\sin \left(\theta_{1}\right)=\frac{1}{2}

\newline

What is the value of

\cos \left(\theta_{1}\right)

\newline

Express your answer exactly.

\newline

\cos \left(\theta_{1}\right)=

Use Pythagorean Identity: We know that $\sin(\theta_{1}) = \frac{1}{2}$ and $\theta_{1}$ is in Quadrant I. Use the Pythagorean identity $\sin^{2}(\theta) + \cos^{2}(\theta) = 1$ to find $\cos(\theta_{1})$ . Substitute $\frac{1}{2}$ for $\sin(\theta_{1})$ in $\sin^{2}(\theta_{1}) + \cos^{2}(\theta_{1}) = 1$ . $\left(\frac{1}{2}\right)^{2} + \cos^{2}(\theta_{1}) = 1$ .
Substitute and Simplify: Simplify $(\frac{1}{2})^2 + \cos^2(\theta_{1}) = 1$ to find the value of $\cos(\theta_{1})$ . $\frac{1}{4} + \cos^2(\theta_{1}) = 1$ $\cos^2(\theta_{1}) = 1 - \frac{1}{4}$ $\cos^2(\theta_{1}) = \frac{3}{4}$
Take Square Root: Since we are looking for $\cos(\theta_{1})$ , we take the square root of both sides. $\newline$ $\cos(\theta_{1}) = \pm\sqrt{\frac{3}{4}}$ $\newline$ $\cos(\theta_{1}) = \pm\frac{\sqrt{3}}{2}$
Determine Sign and Final Answer: Determine the sign of $\cos(\theta_{1})$ based on the quadrant in which $\theta_{1}$ is located. $\newline$ Since $\theta_{1}$ is in Quadrant I, where both sine and cosine are positive, we choose the positive value for $\cos(\theta_{1})$ . $\newline$ $\cos(\theta_{1}) = \sqrt{3}/2$