Find the Exact Value csc(-3pi)

Problem

csc(−3*π)

Solution

1. Identify the definition of the cosecant function in terms of the sine function.

csc(θ)=1/sin(θ)

2. Substitute the given angle into the sine function.

sin(−3*π)

3. Determine the value of the sine function at the given angle using its periodicity. Since sin(θ) has a period of 2*π we can add 4*π to the angle.

−3*π+4*π=π

sin(−3*π)=sin(π)

4. Evaluate the sine of π

sin(π)=0

5. Analyze the cosecant function at this point. Since the sine of the angle is zero, the cosecant is the reciprocal of zero.

csc(−3*π)=1/0

6. Conclude that the value is undefined because division by zero is not possible.

Final Answer

csc(−3*π)=Undefined

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