Find the Exact Value sin((9pi)/7)

Problem

sin((9*π)/7)

Solution

1. Identify the angle and its position on the unit circle. The angle (9*π)/7 is in the third quadrant because π<(9*π)/7<(3*π)/2

2. Determine the reference angle by subtracting π from the given angle.

(θ_ref)=(9*π)/7−π

(θ_ref)=(2*π)/7

3. Apply the quadrant rule for the sine function. In the third quadrant, the sine value is negative.

sin((9*π)/7)=−sin((2*π)/7)

4. Conclude that since (2*π)/7 is not a standard angle (like π/6 or π/4, the exact value is expressed in terms of the sine function.

Final Answer

sin((9*π)/7)=−sin((2*π)/7)

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