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(5π4)

Solution

Rewrite the angles for 5π4

tan(5π4) = tan(π + π4)

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

tan(π + π4) = tan(π4) = 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