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

Problem

tan((19*π)/4)

Solution

1. Reduce the angle by subtracting multiples of the period of the tangent function, which is π

2. Calculate how many full periods are in the fraction by dividing 19 by 4

19÷4=4* with a remainder of *3

3. Rewrite the expression using the identity tan(θ+n*π)=tan(θ)

(19*π)/4=(16*π)/4+(3*π)/4=4*π+(3*π)/4

4. Simplify the expression to its equivalent value within the interval [0,π)

tan((19*π)/4)=tan((3*π)/4)

5. Evaluate the tangent of the reference angle. The angle (3*π)/4 is in the second quadrant where tangent is negative.

tan((3*π)/4)=−1

Final Answer

tan((19*π)/4)=−1

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