Find the Exact Value tan((11pi)/4)

Problem

tan((11*π)/4)

Solution

1. Identify the angle and find its coterminal equivalent within the standard interval [0,2*π) by subtracting multiples of 2*π

(11*π)/4−2*π=(11*π)/4−(8*π)/4

(11*π)/4−(8*π)/4=(3*π)/4

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

3. Find the reference angle θ′ for (3*π)/4 in Quadrant II.

θ′=π−(3*π)/4

θ′=π/4

4. Apply the trigonometric function to the reference angle and determine the sign based on the quadrant. In Quadrant II, the tangent function is negative.

tan((3*π)/4)=−tan(π/4)

5. Substitute the known exact value for tan(π/4) which is 1

−tan(π/4)=−1

Final Answer

tan((11*π)/4)=−1

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