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

## Related Topics

- How to Solve Angles and Angle Measure
- How to Solve Coterminal Angles and Reference Angles
- How to Find Missing Sides and Angles of a Right Triangle
- How to Solve Trig Ratios of General Angles

## 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 \(cos180^\circ=-1,cos90^\circ =0\)

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.

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

### Download Evaluating Trigonometric Function Worksheet

- \(\color{blue}{1}\)
- \(\color{blue}{1}\)
- \(\color{blue}{\sqrt{3}}\)
- \(\color{blue}{\frac{1}{2}}\)
- \(\color{blue}{- \sqrt{3} }\)
- \(\color{blue}{\frac{ \sqrt{3} }{3}}\)