Evaluate csc((7pi)/4)

Problem

csc((7*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. Determine the reference angle for (7*π)/4 by finding its distance from the nearest multiple of π on the x-axis.

Reference Angle=2*π−(7*π)/4=π/4

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

4. Evaluate the sine of the angle, noting that sine is negative in Quadrant IV.

sin((7*π)/4)=−sin(π/4)

sin((7*π)/4)=−√(,2)/2

5. Calculate the cosecant by taking the reciprocal of the sine value.

csc((7*π)/4)=1/(−√(,2)/2)

csc((7*π)/4)=−2/√(,2)

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

csc((7*π)/4)=−(2√(,2))/2

csc((7*π)/4)=−√(,2)

Final Answer

csc((7*π)/4)=−√(,2)

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