What are the exact values of the six trigonometric functions for an angle of 120 degrees?

Finding the Exact Values of the Six Trigonometric Functions for 120 Degrees

The six trigonometric functions include sine, cosine, tangent, cosecant, secant, and cotangent. To find their exact values for an angle of 120 degrees, we can use the unit circle concept and reference angles.

Step 1: Identify the Reference Angle

The angle of 120 degrees is in the second quadrant of the unit circle. To find the reference angle, subtract 120 degrees from 180 degrees:

Reference angle = 180° – 120° = 60°

Step 2: Determine the Values of the Functions

Each trigonometric function can be expressed in terms of the sine and cosine of the reference angle:

• Sine:
  Sine is positive in the second quadrant. Therefore:
  
  sin(120°) = sin(60°) = √3/2
• Cosine:
  Cosine is negative in the second quadrant. Thus:
  
  cos(120°) = -cos(60°) = -1/2
• Tangent:
  Tangent can be calculated as the ratio of sine to cosine. Since sine is positive and cosine is negative:
  
  tan(120°) = sin(120°) / cos(120°) = (√3/2) / (-1/2) = -√3

Step 3: Calculate the Reciprocal Functions

• Cosecant:
  The cosecant function is the reciprocal of sine:
  
  csc(120°) = 1/sin(120°) = 2/√3 (or rationalized as (2√3)/3)
• Secant:
  The secant function is the reciprocal of cosine:
  
  sec(120°) = 1/cos(120°) = -2
• Cotangent:
  The cotangent function is the reciprocal of tangent:
  
  cot(120°) = 1/tan(120°) = -1/√3 (or rationalized as (-√3)/3)

Summary of the Values

By following these steps, you can easily find the exact values of the six trigonometric functions for the angle of 120 degrees!