Find the Exact Value sin((19pi)/4)

Problem

sin((19*π)/4)

Solution

1. Find the coterminal angle by subtracting multiples of 2*π (which is (8*π)/4 from the argument until the angle is within the interval [0,2*π)

(19*π)/4−2*π=(19*π)/4−(8*π)/4=(11*π)/4

(11*π)/4−2*π=(11*π)/4−(8*π)/4=(3*π)/4

2. Identify the quadrant for the angle (3*π)/4 Since π/2<(3*π)/4<π the angle lies in Quadrant II.

3. Determine the sign of the sine function in Quadrant II. In this quadrant, the sine value is positive.

4. Find the reference angle for (3*π)/4 by subtracting the angle from π

π−(3*π)/4=π/4

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

sin(π/4)=√(,2)/2

Final Answer

sin((19*π)/4)=√(,2)/2

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