How to Evaluate Trigonometric Function

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.

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

Download Evaluating Trigonometric Function Worksheet

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

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