Use the given function value(s), and trigonometric identities (including the cofunction identities), to find the indicated trigonometric functions.

A) and
B) and
C) and
D) and
E) and

Question

Use the given function value(s), and trigonometric identities (including the cofunction identities), to find the indicated trigonometric functions. ​
​  ​  ​
A)​ and 
B)​ and 
C) and 
D)​ and 
E)​ and

Quizplus · Accepted Answer

Using the given function value of $f(	heta) = \frac{2}{\sqrt{5}}$, we can use the Pythagorean identity to find $\cos(	heta)$ and $\sin(	heta)$.
$$\cos^2 (	heta) + \sin^2 (	heta) = 1$$
$$\frac{4}{5} + \sin^2 (	heta) = 1$$
$$\sin^2 (	heta) = \frac{1}{5}$$
$$\sin (	heta) = \pm \frac{1}{\sqrt{5}}$$
Since $\frac{2}{\sqrt{5}} > 0$, we know that $\sin(	heta) > 0$, so we have:
$$\sin (	heta) = \frac{1}{\sqrt{5}}$$
Using the cofunction identity, we have:
$$\cos(	heta) = \sin\left(\frac{\pi}{2}-	hetaight) = \cos\left(	heta + \frac{\pi}{2}ight)$$
$$\cos(	heta) = \cos\left(	heta + \frac{\pi}{2}ight) = \frac{2}{\sqrt{5}}$$
Using these values, we can find the remaining trigonometric functions: 
$$	an(	heta) = \frac{\sin(	heta)}{\cos(	heta)} = \frac{1/\sqrt{5}}{2/\sqrt{5}} = \frac{1}{2}$$
$$\cot(	heta) = \frac{1}{	an(	heta)} = \frac{2}{1} = 2$$
$$\csc(	heta) = \frac{1}{\sin(	heta)} = \sqrt{5}$$
$$\sec(	heta) = \frac{1}{\cos(	heta)} = \frac{\sqrt{5}}{2}$$
Therefore, the answer is choice C.

Use the Given Function Value(s), and Trigonometric Identities (Including the Cofunction