# How do you use csctheta=4 to find costheta?

Jan 15, 2017

$\cos \theta = 0.968245836$

#### Explanation:

$\cos e c \theta = 4$

the opposite of $\cos e c \theta = \sin \theta$

$\sin \theta = 0.25$

theta =14°.47751219

= 14°28'39"

$\theta$ lies in the first quadrant where sin and cos= +

$\cos \theta = 0.968245836$

Jan 15, 2017

Use the identities csc(theta) = 1/sin(theta) and cos(theta) = +-sqrt(1 - sin^2(theta)

#### Explanation:

Given: $\csc \left(\theta\right) = 4$ Find $\cos \left(\theta\right)$

Use the identity $\csc \left(\theta\right) = \frac{1}{\sin} \left(\theta\right)$:

$\frac{1}{\sin} \left(\theta\right) = 4$

$\sin \left(\theta\right) = \frac{1}{4}$

Substitute ${\left(\frac{1}{4}\right)}^{2}$ for ${\sin}^{2} \left(\theta\right)$ into the identity: cos(theta) = +-sqrt(1 - sin^2(theta):

$\cos \left(\theta\right) = \pm \sqrt{1 - {\left(\frac{1}{4}\right)}^{2}}$

$\cos \left(\theta\right) = \pm \sqrt{1 - \frac{1}{16}}$

$\cos \left(\theta\right) = \pm \sqrt{\frac{15}{16}}$

$\cos \left(\theta\right) = \pm \frac{\sqrt{15}}{4}$

Because we are not given any clue to whether $\theta$ is in the first or second quadrant, we cannot determine whether the cosine is positive or negative.