QuestionFind $\cos \theta$ if $\sin \theta=\frac{4}{5}$ and $\theta$ is in quadrant II. Simplify your answer.

Studdy Solution

STEP 1

Assumptions1. $\sin \theta=\frac{4}{5}$ . $\theta$ is in quadrant II

STEP 2

We know that $\cos^2 \theta + \sin^2 \theta =1$ . This is the Pythagorean identity in trigonometry.

STEP 3

We can rearrange the Pythagorean identity to solve for $\cos \theta$ .
$\cos^2 \theta =1 - \sin^2 \theta$

STEP 4

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

STEP 5

implify the equation.
$\cos^2 \theta =1 - \frac{16}{25}$

STEP 6

Subtract the fractions.
$\cos^2 \theta = \frac{9}{25}$

STEP 7

Take the square root of both sides to solve for $\cos \theta$ . Remember that the square root of a number can be either positive or negative.
$\cos \theta = \pm \sqrt{\frac{9}{25}}$

STEP 8

implify the square root.
$\cos \theta = \pm \frac{3}{5}$

STEP 9

Since $\theta$ is in quadrant II, and cosine is negative in quadrant II, we choose the negative root.
$\cos \theta = -\frac{3}{5}$ So, the exact value of $\cos \theta$ is $-\frac{3}{5}$ .

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