Find the Exact Value 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. Substitute the sine value back into the reciprocal identity to find the cosecant.

csc((7*π)/4)=1/(−√(,2)/2)

6. Simplify the expression by rationalizing the denominator.

csc((7*π)/4)=−2/√(,2)

csc((7*π)/4)=−√(,2)

Final Answer

csc((7*π)/4)=−√(,2)

Want more problems? Check here!