Find the Exact Value sin((-16pi)/3)

Problem

sin(−(16*π)/3)

Solution

1. Use the odd function property of sine, which states that sin(−θ)=−sin(θ)

sin(−(16*π)/3)=−sin((16*π)/3)

2. Find the coterminal angle by subtracting multiples of 2*π (which is (6*π)/3 from (16*π)/3 until the angle is within the interval [0,2*π)

(16*π)/3−2*π=(16*π)/3−(6*π)/3=(10*π)/3

(10*π)/3−2*π=(10*π)/3−(6*π)/3=(4*π)/3

3. Determine the reference angle for (4*π)/3 which is located in Quadrant III.

(θ_ref)=(4*π)/3−π=π/3

4. Evaluate the sine of the reference angle.

sin(π/3)=√(,3)/2

5. Apply the quadrant sign for Quadrant III, where sine is negative.

sin((4*π)/3)=−√(,3)/2

6. Substitute back into the expression from step 1.

−sin((16*π)/3)=−(−√(,3)/2)=√(,3)/2

Final Answer

sin(−(16*π)/3)=√(,3)/2

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