Finding the Exact Values of Trigonometric Functions
To find the exact values of the given trigonometric functions, let’s calculate each one step by step:
1. Calculate sin(4π/3)
The angle 4π/3 radians is in the third quadrant. To find sin(4π/3):
- The reference angle is: 4π/3 – π = π/3.
- In the third quadrant, the sine value is negative.
- Thus, sin(4π/3) = -sin(π/3) = -√3/2.
2. Calculate sec(7π/3)
The angle 7π/3 can be converted to an equivalent angle:
- Subtract 2π (or 6π/3) from 7π/3 to get 7π/3 – 6π/3 = π/3.
- Now, find sec(π/3).
Since sec(x) = 1/cos(x), we have:
- sec(π/3) = 1/cos(π/3) = 1/(1/2) = 2.
3. Calculate cot(2π/3)
The angle 2π/3 is also in the second quadrant:
- The reference angle here is: 2π/3 – π = π/3.
- In the second quadrant, the cosine value is negative.
- cot(2π/3) = 1/tan(2π/3) = -1/tan(π/3) = -1/(√3) = -√3/3.
Final Answers:
- sin(4π/3) = -√3/2
- sec(7π/3) = 2
- cot(2π/3) = -√3/3