Question

What is tan 0 when csc 0= 2/3

Answer 1

`tan(\theta) = (√(11))/(11)`

Explanation:

Cosecant:

The cosecant is one divided by the sine. Thus:

`\csc{\theta} = (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}} = (1)/(2√(3)) * (√(3))/(√(3)) = (√(3))/(6)`

`\sin^(2){\theta} + \cos^(2){\theta} = 1`

`\cos^(2){\theta} = 1 - \sin^(2){\theta}`

`\cos^(2){\theta} = 1 - ((√(3))/(6))^2`

`\cos^(2){\theta} = 1 - (3)/(36)`

`\cos^(2){\theta} = (33)/(36)`

First quadrant, so the cosine is positive. Then

`\cos^(2){\theta} = \sqrt{(33)/(36)} = (√(33))/(6)`

Tangent:

Sine divided by cosine. So

`tan(\theta) = (sin(\theta))/(cos(\theta)) = ((√(3))/(6))/((√(33))/(6)) = (√(3))/(√(33)) = (√(3))/(√(3)√(11)) = (1)/(√(11)) * (√(11))/(√(11)) = (√(11))/(11)`

The answer is:

`tan(\theta) = (√(11))/(11)`