Question

If csctheta=11/4 & tan Thea< 0 Use the fundamental identities to evaluate cottheta.

Answer 1

`\cot{\theta} = -(√(105))/(11)`

Explanation:

Fundamental identities:

These following trigonometric identities are used to solve this question:

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

`\csc{\theta} = (1)/(sin(\theta))`

`\cot{\theta} = (cos(\theta))/(sin(\theta))`

With the cossecant, we can find the sine. So:

`\csc{\theta} = (1)/(sin(\theta))`

`(11)/(4) = (1)/(sin(\theta))`

`11sin(\theta) = 4`

`sin(\theta) = (4)/(11)`

With the sine, we find the cosine.

Since the tangent is negative, the cosine is negative. So

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

`((4)/(11))^(2) + \cos^(2){\theta} = 1`

`(16)/(121) + \cos^(2){\theta} = (121)/(121)`

`\cos^(2){\theta} = (105)/(121)`

`cos(\theta) = -(√(105))/(11)`

Cotangent:

`\cot{\theta} = (cos(\theta))/(sin(\theta))`

`\cot{\theta} = (-(√(105))/(11))/((4)/(11))`

`\cot{\theta} = -(√(105))/(11)`