Find the Exact Value tan(3pi)

Problem

tan(3*π)

Solution

1. Identify the period of the tangent function. The tangent function tan(θ) has a period of π meaning tan(θ+n*π)=tan(θ) for any integer n

2. Simplify the angle using the periodicity. Since 3*π=0+3*π the value of tan(3*π) is equivalent to tan(0)

3. Apply the definition of tangent in terms of sine and cosine.

tan(3*π)=sin(3*π)/cos(3*π)

4. Evaluate the trigonometric functions at the given angle. On the unit circle, the angle 3*π (which is 2*π+π corresponds to the point (−1,0)

sin(3*π)=0

cos(3*π)=−1

5. Calculate the final ratio.

tan(3*π)=0/(−1)=0

Final Answer

tan(3*π)=0

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