The angle $\theta_{1}$ is located in Quadrant $\mathrm{I}$, and $\sin \left(\theta_{1}\right)=\frac{11}{61}$.
What is the value of $\cos \left(\theta_{1}\right)$ ?
Express your answer exactly.
$$
\cos \left(\theta_{1}\right)=
$$
$\square$

Q: The angle $\theta_{1}$ is located in Quadrant $\mathrm{I}$, and $\sin \left(\theta_{1}\right)=\frac{11}{61}$. What is the value of $\cos \left(\theta_{1}\right)$ ? Express your answer exactly. \[ \cos \left(\theta_{1}\right)= \] $\square$

∻Solution by Steps∻ ‖ step 1 ⋮ Given that $\sin(\theta_1) = \frac{11}{61}$ and $\theta_1$ is in Quadrant I. ‖ ‖ step 2 ⋮ Use the Pythagorean identity: $\sin^2(\theta) + \cos^2(\theta) = 1$. ‖ ‖ step 3 ⋮ Substitute $\sin(\theta_1)$ into the identity: $\left(\frac{11}{61}\right)^2 + \cos^2(\theta_1) = 1$. ‖ ‖ step 4 ⋮ Calculate $\sin^2(\theta_1)$: $\left(\frac{11}{61}\right)^2 = \frac{121}{3721}$. ‖ ‖ step 5 ⋮ Solve for $\cos^2(\theta_1)$: $\cos^2(\theta_1) = 1 - \frac{121}{3721}$. ‖ ‖ step 6 ⋮ Simplify the expression: $\cos^2(\theta_1) = \frac{3721}{3721} - \frac{121}{3721} = \frac{3600}{3721}$. ‖ ‖ step 7 ⋮ Take the square root of $\cos^2(\theta_1)$ to find $\cos(\theta_1)$: $\cos(\theta_1) = \sqrt{\frac{3600}{3721}}$. ‖ ‖ step 8 ⋮ Since $\theta_1$ is in Quadrant I, $\cos(\theta_1)$ is positive: $\cos(\theta_1) = \frac{60}{61}$. ‖ ∻[1] Answer∻ ⚹ $\cos(\theta_1) = \frac{60}{61}$ ⚹ ∻Key Concept∻ ⚹Pythagorean Identity in Trigonometry⚹ ∻Explanation∻ ⚹The Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ is used to find the cosine of an angle when the sine is known, especially when the angle is in the first quadrant where both sine and cosine are positive.⚹◊What trigonometric identity can be used to relate sine and cosine in Quadrant I?⍭ Generate me a similar question◊

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Given that $\sin( heta_1) = \frac{11}{61}$ and $ heta_1$ is in Quadrant I. ‖ ‖ step 2 ⋮ Use the Pythagorean identity: $\sin^2( heta) + \cos^2( heta) = 1$. ‖ ‖ step 3 ⋮ Substitute $\sin( heta_1)$ into the identity: $\left(\frac{11}{61} ight)^2 + \cos^2( heta_1) = 1$. ‖ ‖ step 4 ⋮ Calculate $\sin^2( heta_1)$: $\left(\frac{11}{61} ight)^2 = \frac{121}{3721}$. ‖ ‖ step 5 ⋮ Solve for $\cos^2( heta_1)$: $\cos^2( heta_1) = 1 - \frac{121}{3721}$. ‖ ‖ step 6 ⋮ Simplify the expression: $\cos^2( heta_1) = \frac{3721}{3721} - \frac{121}{3721} = \frac{3600}{3721}$. ‖ ‖ step 7 ⋮ Take the square root of $\cos^2( heta_1)$ to find $\cos( heta_1)$: $\cos( heta_1) = \sqrt{\frac{3600}{3721}}$. ‖ ‖ step 8 ⋮ Since $ heta_1$ is in Quadrant I, $\cos( heta_1)$ is positive: $\cos( heta_1) = \frac{60}{61}$. ‖ ∻[1] Answer∻ ⚹ $\cos( heta_1) = \frac{60}{61}$ ⚹ ∻Key Concept∻ ⚹Pythagorean Identity in Trigonometry⚹ ∻Explanation∻ ⚹The Pythagorean identity $\sin^2( heta) + \cos^2( heta) = 1$ is used to find the cosine of an angle when the sine is known, especially when the angle is in the first quadrant where both sine and cosine are positive.⚹◊What trigonometric identity can be used to relate sine and cosine in Quadrant I?⍭ Generate me a similar question◊

Sia