How to Evaluate Trigonometric Function? (+FREE Worksheet!)

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

How to Evaluate Trigonometric Function? (+FREE Worksheet!)

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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, \(cos\) \(270^\circ=0\)

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 \(cos\) 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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