Find the Exact Value tan((13pi)/4)

Problem

tan((13*π)/4)

Solution

1. Identify the angle and determine its position on the unit circle by finding a coterminal angle within the interval [0,2*π)

2. Subtract multiples of 2*π (which is (8*π)/4 from the given angle.

(13*π)/4−(8*π)/4=(5*π)/4

3. Determine the quadrant for the angle (5*π)/4 Since π<(5*π)/4<(3*π)/2 the angle is in Quadrant III.

4. Find the reference angle for (5*π)/4 by subtracting π

(5*π)/4−π=π/4

5. Apply the tangent function properties. In Quadrant III, the tangent function is positive.

tan((5*π)/4)=tan(π/4)

6. Evaluate the exact value using the known trigonometric value for π/4

tan(π/4)=1

Final Answer

tan((13*π)/4)=1

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