# Assume that theta is an acute angle in a right triangle satisfying the given condition. Evaluate the remaining trigonometric functions ? sin theta= 4/11

## Find $\cos \theta =$ $\tan \theta =$ $\csc \theta =$ $\sec \theta =$ $\cot \theta =$

May 13, 2018

See explanation.

#### Explanation:

First we can notice that all the values are positive because the angle is accute (i.e. it is located in $Q 1$).

To calculate $\cos \theta$ we can use the identity saying that for any angle we have:

## ${\sin}^{2} \theta + {\cos}^{2} \theta = 1$

If we use the given value we get:

## ${\left(\frac{4}{11}\right)}^{2} + {\cos}^{2} \theta = 1$

${\cos}^{2} \theta = 1 - {\left(\frac{4}{11}\right)}^{2}$

${\cos}^{2} \theta = 1 - \frac{16}{121} = \frac{105}{121}$

$\cos \theta = \frac{\sqrt{105}}{11}$

Now having $\sin \theta$ and $\cos \theta$ we can calculate the remaining functions:

## $\tan \theta = \sin \frac{\theta}{\cos} \theta$ 

$\tan \theta = \frac{4}{11} \div \frac{\sqrt{105}}{11} = \frac{4}{11} \cdot \frac{11}{\sqrt{105}} = \frac{4}{\sqrt{105}} = \frac{4 \sqrt{105}}{105}$

$\cot \theta = \frac{1}{\tan} \theta = \frac{\sqrt{105}}{4}$

$\sec \theta = \frac{1}{\cos} \theta = \frac{11}{\sqrt{105}} = \frac{11 \sqrt{105}}{105}$

$\csc \theta = \frac{1}{\sin} \theta = \frac{11}{4} = 2 \frac{3}{4}$