Question

What is tan 0 when csc 0= 2/3

Answer 1

Answer:

$\tan{\theta} = \frac{\sqrt{11}}{11}$

Step-by-step explanation:

Cosecant:

The cosecant is one divided by the sine. Thus:

$\csc{\theta} = \frac{1}{\sin{\theta}}$

Tangent is sine divided by cosine, so we first find the sine, then the cosine, to find the tangent.

Sine and cosine:

$\sin{\theta} = \frac{1}{\csc{\theta}} = \frac{1}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{6}$

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

$\cos^{2}{\theta} = 1 - \sin^{2}{\theta}$

$\cos^{2}{\theta} = 1 - (\frac{\sqrt{3}}{6})^2$

$\cos^{2}{\theta} = 1 - \frac{3}{36}$

$\cos^{2}{\theta} = \frac{33}{36}$

First quadrant, so the cosine is positive. Then

$\cos^{2}{\theta} = \sqrt{\frac{33}{36}} = \frac{\sqrt{33}}{6}$

Tangent:

Sine divided by cosine. So

$\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} = \frac{\frac{\sqrt{3}}{6}}{\frac{\sqrt{33}}{6}} = \frac{\sqrt{3}}{\sqrt{33}} = \frac{\sqrt{3}}{\sqrt{3}\sqrt{11}} = \frac{1}{\sqrt{11}} \times \frac{\sqrt{11}}{\sqrt{11}} = \frac{\sqrt{11}}{11}$

The answer is:

$\tan{\theta} = \frac{\sqrt{11}}{11}$