QuestionFind $\cos \theta$ if $\sin \theta=\frac{4}{5}$ and $\theta$ is in quadrant I.

Studdy Solution

STEP 1

Assumptions1. The value of $\sin \theta$ is given as $\frac{4}{5}$ . $\theta$ is in quadrant I3. We are asked to find the exact value of $\cos \theta$
4. We will use the Pythagorean identity $\sin^ \theta + \cos^ \theta =1$

STEP 2

First, we need to use the Pythagorean identity to express $\cos \theta$ in terms of $\sin \theta$ .
$\cos^2 \theta =1 - \sin^2 \theta$

STEP 3

Now, plug in the given value for $\sin \theta$ to calculate $\cos^2 \theta$ .
$\cos^2 \theta =1 - \left(\frac{}{5}\right)^2$

STEP 4

Calculate the square of $\frac{4}{}$ .
$\left(\frac{4}{}\right)^2 = \frac{16}{25}$

STEP 5

Substitute the value of $\left(\frac{4}{5}\right)^2$ into the equation for $\cos^2 \theta$ .
$\cos^2 \theta =1 - \frac{16}{25}$

STEP 6

Subtract $\frac{16}{25}$ from1 to find $\cos^2 \theta$ .
$\cos^2 \theta = \frac{9}{25}$

STEP 7

Now that we have the value of $\cos^2 \theta$ , we can find the value of $\cos \theta$ by taking the square root of both sides of the equation. Remember that the square root of a number can be positive or negative, but since $\theta$ is in quadrant I where cosine is positive, we only take the positive root.
$\cos \theta = \sqrt{\cos^2 \theta}$

STEP 8

Plug in the value for $\cos^2 \theta$ to calculate $\cos \theta$ .
$\cos \theta = \sqrt{\frac{}{25}}$

STEP 9

Calculate the square root of $\frac{9}{25}$ .
$\cos \theta = \frac{3}{5}$ So, the exact value of $\cos \theta$ is $\frac{3}{5}$ .

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