Question

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

Answer 1

Answer:

$\cot{\theta} = -\frac{\sqrt{105}}{11}$

Step-by-step explanation:

Fundamental identities:

These following trigonometric identities are used to solve this question:

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

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

$\cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}}$

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

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

$\frac{11}{4} = \frac{1}{\sin{\theta}}$

$11\sin{\theta} = 4$

$\sin{\theta} = \frac{4}{11}$

With the sine, we find the cosine.

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

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

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

$\frac{16}{121} + \cos^{2}{\theta} = \frac{121}{121}$

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

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

Cotangent:

$\cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}}$

$\cot{\theta} = \frac{-\frac{\sqrt{105}}{11}}{\frac{4}{11}}$

$\cot{\theta} = -\frac{\sqrt{105}}{11}$