How to Evaluate Each Trigonometric Function

How to Evaluate Each Trigonometric Function 

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Trigonometric Functions

Trigonometric functions -also referred to as circular functions- are the functions of a triangle's angle. This means that these trig functions provide the relationship between the angles and sides of a triangle. Sine, cosine, tangent, cotangent, secant, and cosecant are the basic trigonometric functions.

How to Evaluate Trigonometric Functions Step by Step

  • Determine 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 reference angle's trigonometric function.

Example

Evaluate \(tan(\frac{5π}{4})\)

Solution

Rewrite the angles for \(\frac{5π}{4}\)

\(tan(\frac{5π}{4}) \ = \ tan(π \ + \ \frac{π}{4})\)

Utilize the \(tan\) periodicity: \(tan(θ \ + \ nπ) \ = \ tan(θ)\)

\(tan(π \ + \ \frac{π}{4}) \ = \ tan(\frac{π}{4}) \ = \ 1\)

Additional Angle Identities

  • \(Sin(π \ - \ θ) \ = \ sin(θ)\)
  • \(Cos(π \ - \ θ) \ = \ –cos(θ)\)
  • \(tan(π \ - \ θ) \ = \ – tan(θ)\)
  • \(Cosec(π \ - \ θ) \ = \ cosec(θ)\)
  • \(Sec(π \ - \ θ) \ = \ -sec(θ)\)
  • \(Cot(π \ - \ θ) \ = \ -cot(θ)\)

Free printable Worksheets

Exercises for Evaluating Each Trigonometric Function

1) \(sin \ 120° \ =\)

2) \(tan \ 150° \ =\)

3) \(cot \ 330° \ =\)

4) \(tan \ 300° \ =\)

5) \(cos \ 225° \ =\)

6) \(sin \ \frac{3π}{2} \ =\)

7) \(cot \ \frac{11π}{6} \ =\)

8) \(sin \ \frac{2π}{3} \ =\)

9) \(cos \ \frac{7π}{6} \ =\)

10) \(tan \ \frac{9π}{4} \ =\)

 

1) \(sin \ 120° \ =\)

\(\color{red}{sin \ 120° \ = \ sin \ (180° \ - \ 60°) \ = \ sin \ 60° \ = \ \frac{\sqrt{3}}{2}}\)

2) \(tan \ 150° \ =\)

\(\color{red}{tan \ 150° \ = \ tan \ (180° \ - \ 30°) \ = \ -tan \ 30° \ = \ -\frac{\sqrt{3}}{3}}\)

3) \(cot \ 330° \ =\)

\(\color{red}{cot \ 330° \ = \ cot \ (360° \ - \ 30°) \ = \ -cot \ 30° \ = \ -\sqrt{3}}\)

4) \(tan \ 300° \ =\)

\(\color{red}{tan \ 300° \ = \ tan \ (300° \ - \ 60°) \ = \ -tan \ 60° \ = \ -\sqrt{3}}\)

5) \(cos \ 225° \ =\)

\(\color{red}{cos \ 225° \ = \ cos \ (180° \ + \ 45°) \ = \ -cos \ 45° \ = \ -\frac{\sqrt{2}}{2}}\)

6) \(sin \ \frac{3π}{2} \ =\)

\(\color{red}{sin \ \frac{3π}{2} \ = \ sin \ (π \ + \ \frac{π}{2}) \ = \ -sin \ \frac{π}{2} \ = -1}\)

7) \(cot \ \frac{11π}{6} \ =\)

\(\color{red}{cot \ \frac{11π}{6} \ = \ cot \ (2π \ - \ \frac{π}{6}) \ = \ -cot \ \frac{π}{6} \ = -\sqrt{3}}\)

8) \(sin \ \frac{2π}{3} \ =\)

\(\color{red}{sin \ \frac{2π}{3} \ = \ sin \ (π \ - \ \frac{π}{3}) \ = \ sin \ \frac{π}{3} \ = \frac{\sqrt{3}}{2}}\)

9) \(cos \ \frac{7π}{6} \ =\)

\(\color{red}{cos \ \frac{7π}{6} \ = \ cos \ (π \ + \ \frac{π}{6}) \ = \ -cos \ \frac{π}{6} \ = -\frac{\sqrt{3}}{2}}\)

10) \(tan \ \frac{9π}{4} \ =\)

\(\color{red}{tan \ \frac{9π}{4} \ = \ tan \ (2π \ + \ \frac{π}{4}) \ = \ tan \ \frac{π}{4} \ = \frac{\sqrt{2}}{2}}\)

Evaluating Each Trigonometric Function Practice Quiz