Evaluating Trigonometric Function

Evaluating 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.

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\(⁡(\frac{1}{3} π)=\sqrt{3} \)

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 \(90^\circ=0\)

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 \(225^\circ=-\frac{\sqrt{2}}{2}\)

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

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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