Find the Exact Value sin(-(23pi)/3)

Problem

sin(−(23*π)/3)

Solution

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

−sin((23*π)/3)

2. Find the coterminal angle by dividing the numerator by the denominator to identify the number of full rotations.

(23*π)/3=7*π+(2*π)/3

3. Subtract multiples of 2*π to find an angle within the interval [0,2*π)

(23*π)/3−3*(2*π)=(23*π)/3−(18*π)/3

(23*π)/3≡(5*π)/3

4. Substitute the coterminal angle back into the expression.

−sin((5*π)/3)

5. Determine the reference angle for (5*π)/3 which is in the fourth quadrant.

Reference Angle=2*π−(5*π)/3=π/3

6. Apply the sine value for the reference angle, noting that sine is negative in the fourth quadrant.

sin((5*π)/3)=−sin(π/3)

sin((5*π)/3)=−√(,3)/2

7. Simplify the final expression by accounting for the negative sign from step 1.

−(−√(,3)/2)=√(,3)/2

Final Answer

sin(−(23*π)/3)=√(,3)/2

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