Find the Exact Value sin((4pi)/3)

Problem

sin((4*π)/3)

Solution

1. Identify the quadrant of the angle (4*π)/3 Since π<(4*π)/3<(3*π)/2 the angle is in the third quadrant.

2. Determine the reference angle. For an angle θ in the third quadrant, the reference angle θ′ is calculated as θ−π

θ′=(4*π)/3−π=π/3

3. Apply the sign for the sine function in the third quadrant. In the third quadrant, the sine of an angle is negative.

sin((4*π)/3)=−sin(π/3)

4. Evaluate the sine of the reference angle using known trigonometric values.

sin(π/3)=√(,3)/2

5. Combine the sign and the value to find the final result.

sin((4*π)/3)=−√(,3)/2

Final Answer

sin((4*π)/3)=−√(,3)/2

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