Find the Exact Value tan((15pi)/4)

Problem

tan((15*π)/4)

Solution

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

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

(15*π)/4−2*π=(15*π)/4−(8*π)/4

(15*π)/4−(8*π)/4=(7*π)/4

3. Determine the quadrant of the coterminal angle (7*π)/4 Since (3*π)/2<(7*π)/4<2*π the angle is in Quadrant IV.

4. Find the reference angle (θ_R) for an angle in Quadrant IV using the formula (θ_R)=2*π−θ

(θ_R)=2*π−(7*π)/4

(θ_R)=π/4

5. Apply the tangent function to the reference angle and determine the sign based on the quadrant. In Quadrant IV, tangent is negative.

tan((7*π)/4)=−tan(π/4)

6. Substitute the known exact value tan(π/4)=1

−tan(π/4)=−1

Final Answer

tan((15*π)/4)=−1

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