How to Evaluate Trigonometric Function? (+FREE Worksheet!)
How to evaluate trigonometric function, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.
Evaluating Trigonometric Function, Example 1:
Find the exact value of the trigonometric function. \(tan\)\(\frac{4π}{3}\)
Evaluate Trigonometric Function: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Choose the modelUse a right triangle, the unit circle, or a transformed graph.
- Track unitsConvert degrees and radians when needed.
- Use identitiesReplace complicated trig expressions with equivalent simpler ones.
Worked examples
Right-triangle sine
- Sine is opposite over hypotenuse.
- Substitute 5 and 13.
- Leave the ratio simplified.
Unit-circle cosine
- At angle 0, the point is (1, 0).
- Cosine is the x-coordinate.
- Read the x-value.
Try one before moving on
Evaluate Trigonometric Function: pop-up practice
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 \(tan\): \(tan\)\((x+π.k)= tan (x)\)
\(tan\)\((π+\frac{1}{3} π)=\) \(tan\)\((\frac{1}{3} π)=\sqrt{3} \)
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Evaluating Trigonometric Function, Example 2:
Find the exact value of the trigonometric function. \(cos\) \(270^\circ\)
Solution:
Write \(cos\) \((270^\circ)\) as \(cos\) \((180^\circ+90^\circ)\). Recall that \(cos180^\circ=-1,cos 90^\circ =0\)
The reference angle of \(270^\circ\) is \(90^\circ\). Therefore, \(cos\) \(270^\circ=0\)
Evaluating Trigonometric Function, Example 3:
Find the exact value of the 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, \(cos\) \(225^\circ=-\frac{\sqrt{2}}{2}\)
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Evaluating Trigonometric Function, Example 4:
Find the exact value of the 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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