Question

Math

Posted 3 months ago

```
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.
$\cos \left(\theta_{1}\right)=$
$\square$
```

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Answer from Sia

Posted 3 months ago

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.

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