Find the Exact Value csc((3pi)/4)

Problem

csc((3*π)/4)

Solution

1. Identify the relationship between the cosecant function and the sine function using the reciprocal identity.

csc(θ)=1/sin(θ)

2. Substitute the given angle into the identity.

csc((3*π)/4)=1/sin((3*π)/4)

3. Determine the value of the sine function at the angle (3*π)/4 Since this angle is in the second quadrant, the sine value is positive.

sin((3*π)/4)=√(,2)/2

4. Calculate the reciprocal of the sine value to find the cosecant.

csc((3*π)/4)=1/√(,2)/2

5. Simplify the fraction by multiplying by the reciprocal.

csc((3*π)/4)=2/√(,2)

6. Rationalize the denominator by multiplying the numerator and denominator by √(,2)

csc((3*π)/4)=(2√(,2))/2

7. Reduce the expression to its simplest form.

csc((3*π)/4)=√(,2)

Final Answer

csc((3*π)/4)=√(,2)

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