Simplify sin((5pi)/3)

Problem

sin((5*π)/3)

Solution

1. Identify the quadrant of the angle (5*π)/3 Since (3*π)/2<(5*π)/3<2*π the angle is in the fourth quadrant.

2. Determine the reference angle. In the fourth quadrant, the reference angle (θ_R) is calculated as 2*π−θ

(θ_R)=2*π−(5*π)/3

(θ_R)=π/3

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

sin((5*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((5*π)/3)=−√(,3)/2

Final Answer

sin((5*π)/3)=−√(,3)/2

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