# Determine the exact values of the trigonometric functions of the acute angle A, given tan A =3/7 ?

Feb 26, 2018

see explanation

#### Explanation:

Tan A = 3/7 specifies a right triangle with base = 7, height = 3.

(because by definition Tan A = height/base).

The hypotenuse of the triangle is then $\sqrt{{7}^{2} + {3}^{2}} = 7.616$ (rounding).

And with this information, you can calculate the other trig values for the triangle:

$\sin a = \frac{3}{7.616}$

$\cos a = \frac{7}{7.616}$

...and, as a final sanity check, you can use the alternate definition for Tan A:

$\tan a = \frac{\sin a}{\cos a}$

$= \frac{\frac{3}{7.616}}{\frac{7}{7.616}}$

$= \frac{3}{7.616} \cdot \frac{7.616}{7}$

$= \frac{3}{7}$

...which checks out.

I won't calculate sec a, csc a, cot a for you, you should be able to take it from here, since:

$\sec a = \frac{1}{\cos a}$, $\csc a = \frac{1}{\sin a}$, and $\cot a = \frac{1}{\tan a}$

GOOD LUCK

Feb 26, 2018

$\sin \left(A\right) = \frac{3 \sqrt{58}}{58}$
$\csc \left(A\right) = \frac{\sqrt{58}}{3}$
$\cos \left(A\right) = \frac{7 \sqrt{58}}{58}$
$\sec \left(A\right) = \frac{\sqrt{58}}{7}$
$\cot \left(A\right) = \frac{7}{3}$

#### Explanation:

Given: $\tan \left(A\right) = \frac{3}{7} , 0 < A < \frac{\pi}{2}$

Use the identity $\cot \left(A\right) = \frac{1}{\tan} \left(A\right)$

$\cot \left(A\right) = \frac{7}{3}$

Use the identity $1 + {\tan}^{2} \left(A\right) = {\sec}^{2} \left(A\right)$

$1 + {\left(\frac{3}{7}\right)}^{2} = {\sec}^{2} \left(A\right)$

$1 + \frac{9}{49} = {\sec}^{2} \left(A\right)$

${\sec}^{2} \left(A\right) = \frac{58}{49}$

$\sec \left(A\right) = \frac{\sqrt{58}}{7}$

Using the identity $\cos \left(A\right) = \frac{1}{\sec} \left(A\right)$

$\cos \left(A\right) = \frac{7 \sqrt{58}}{58}$

Using the identity $\tan \left(A\right) = \sin \frac{A}{\cos} \left(A\right)$

$\frac{3}{7} = \sin \frac{A}{\frac{7 \sqrt{58}}{58}}$

$\sin \left(A\right) = \frac{3 \sqrt{58}}{58}$

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

$\csc \left(A\right) = \frac{\sqrt{58}}{3}$