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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