Find the Exact Value csc(-45)

Problem

csc(−45)

Solution

1. Apply the odd-even identity for the cosecant function, which states that csc(−θ)=−csc(θ)

csc(−45)=−csc(45)

2. Use the reciprocal identity to express cosecant in terms of sine, where csc(θ)=1/sin(θ)

−csc(45)=−1/sin(45)

3. Substitute the known value for sin(45) which is √(,2)/2

−1/sin(45)=−1/√(,2)/2

4. Simplify the fraction by multiplying by the reciprocal.

−1/√(,2)/2=−2/√(,2)

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

−2/√(,2)⋅√(,2)/√(,2)=−(2√(,2))/2

6. Simplify the expression by canceling the common factor of 2.

−(2√(,2))/2=−√(,2)

Final Answer

csc(−45)=−√(,2)

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