# How to Evaluate Trigonometric Function

Learn how to evaluate trigonometric functions in few simple steps with examples and detailed solutions.

## Step by step guide to Evaluating Trigonometric Function

• Find the reference angle. (It is the smallest angle that you can make from the terminal side of an angle with the $$x$$-axis.)
• Find the trigonometric function of the reference angle.

### Evaluating Trigonometric Function – Example 1:

Find the exact value of trigonometric function. tan$$\frac{4π}{3}$$

Solution:

Rewrite the angles for an $$\frac{4π}{3}$$:
tan $$\frac{4π}{3}=tan (\frac{3π+π}{3})=tan⁡(π+\frac{1}{3} π)$$
Use the periodicity of tan: tan$$(x+π .k)=$$ tan$$(x)$$
tan$$⁡(π+\frac{1}{3} π)=$$ tan$$⁡(\frac{1}{3} π)=\sqrt{3}$$

### Evaluating Trigonometric Function – Example 2:

Find the exact value of trigonometric function. cos $$270^\circ$$

Solution:

Write cos $$(270^\circ)$$ as cos $$(180^\circ+90^\circ)$$. Recall that $$cos⁡180^\circ=-1,cos⁡90^\circ =0$$
The reference angle of $$270^\circ$$ is $$90^\circ$$. Therefore, cos $$90^\circ=0$$

### Evaluating Trigonometric Function – Example 3:

Find the exact value of trigonometric function. cos $$225^\circ$$

Solution:

Write cos $$(225^\circ)$$ as cos $$(180^\circ+45^\circ)$$. Recall that cos$$⁡180^\circ=-1$$,cos$$⁡45^\circ =\frac{\sqrt{2}}{2}$$
$$225^\circ$$ is in the third quadrant and cosine is negative in the quadrant $$3$$. The reference angle of $$225^\circ$$ is $$45^\circ$$. Therefore, cos $$225^\circ=-\frac{\sqrt{2}}{2}$$

### Evaluating Trigonometric Function – Example 4:

Find the exact value of trigonometric function. tan $$\frac{7π}{6}$$

Solution:

Rewrite the angles for tan $$\frac{7π}{6}$$:
tan $$\frac{7π}{6}=$$ tan $$(\frac{6π+π}{6})=tan⁡(π+\frac{1}{6} π)$$
Use the periodicity of tan: tan$$(x+π .k)=$$ tan$$(x)$$
$$tan⁡(π+\frac{1}{6} π)=$$ tan$$⁡(\frac{1}{6} π)=\frac{\sqrt{3}}{3}$$

## Exercises for Evaluating Trigonometric Function

### Find the exact value of each trigonometric function.

• $$\color{blue}{cot \ -495^\circ=}$$
• $$\color{blue}{tan \ 405^\circ=}$$
• $$\color{blue}{cot \ 390^\circ=}$$
• $$\color{blue}{cos \ -300^\circ=}$$
• $$\color{blue}{cot \ -210^\circ=}$$
• $$\color{blue}{tan \ \frac{7π}{6}=}$$

• $$\color{blue}{1}$$
• $$\color{blue}{1}$$
• $$\color{blue}{\sqrt{3}}$$
• $$\color{blue}{\frac{1}{2}}$$
• $$\color{blue}{- \sqrt{3} }$$
• $$\color{blue}{\frac{ \sqrt{3} }{3}}$$

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