$\newline$ The angle $\theta_{1}$ is located in Quadrant $\mathrm{I}$ , and $\sin \left(\theta_{1}\right)=\frac{17}{20}$ . $\newline$ 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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Find trigonometric ratios using a Pythagorean or reciprocal identity

Full solution

\newline

The angle

\theta_{1}

is located in Quadrant

\mathrm{I}

, and

\sin \left(\theta_{1}\right)=\frac{17}{20}

\newline

What is the value of

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

\newline

Express your answer exactly.

\newline

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

Understand Relationship: Understand the relationship between sine and cosine in a right triangle. $\newline$ In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse, while the cosine is the ratio of the length of the adjacent side to the hypotenuse. For an angle $\theta_1$ in the first quadrant, both sine and cosine values are positive. We can use the Pythagorean theorem to find the cosine if we know the sine.
Use Pythagorean Identity: Use the Pythagorean identity to find $\cos(\theta_1)$ . The Pythagorean identity states that $\sin^2(\theta_1) + \cos^2(\theta_1) = 1$ . We can solve for $\cos^2(\theta_1)$ by substituting the given value of $\sin(\theta_1)$ . $\sin^2(\theta_1) + \cos^2(\theta_1) = 1$ $\left(\frac{17}{20}\right)^2 + \cos^2(\theta_1) = 1$
Calculate $\sin^2$ : Calculate $\sin^2(\theta_1)$ . $\left(\frac{17}{20}\right)^2 = \left(\frac{17^2}{20^2}\right) = \frac{289}{400}$
Substitute and Solve: Substitute $\sin^2(\theta_1)$ into the Pythagorean identity and solve for $\cos^2(\theta_1)$ . $\frac{289}{400} + \cos^2(\theta_1) = 1$ $\cos^2(\theta_1) = 1 - \frac{289}{400}$ $\cos^2(\theta_1) = \frac{400}{400} - \frac{289}{400}$ $\cos^2(\theta_1) = \frac{400 - 289}{400}$ $\cos^2(\theta_1) = \frac{111}{400}$
Find $\cos(\theta_1):$ Find the value of $\cos(\theta_1)$ . Since $\theta_1$ is in the first quadrant, $\cos(\theta_1)$ will be positive. Therefore, we take the positive square root of $\cos^2(\theta_1)$ . $\cos(\theta_1) = \sqrt{\frac{111}{400}}$ $\cos(\theta_1) = \frac{\sqrt{111}}{\sqrt{400}}$ $\cos(\theta_1) = \frac{\sqrt{111}}{20}$