Find the Exact Value csc(495)

Problem

csc(495)

Solution

1. Find the coterminal angle by subtracting 360 from the given angle to bring it within the range of 0 to 360

495−360=135

2. Identify the reference angle for 135 which is located in the second quadrant.

180−135=45

3. Determine the sign of the cosecant function in the second quadrant. Since sin(θ) is positive in the second quadrant, csc(θ) is also positive.

csc(135)=csc(45)

4. Apply the reciprocal identity for cosecant, which is the reciprocal of the sine function.

csc(45)=1/sin(45)

5. Substitute the known value for sin(45)=√(,2)/2

csc(45)=1/√(,2)/2

6. Simplify the expression by multiplying by the reciprocal.

1/√(,2)/2=2/√(,2)

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

(2√(,2))/2=√(,2)

Final Answer

csc(495)=√(,2)

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