Find the Exact Value csc(-pi/4)

Problem

csc(−π/4)

Solution

1. Identify the trigonometric identity for the cosecant function, which is the reciprocal of the sine function.

csc(θ)=1/sin(θ)

2. Apply the odd function property of the sine function, which states that sin(−θ)=−sin(θ)

sin(−π/4)=−sin(π/4)

3. Determine the exact value of sin(π/4) from the unit circle or a special right triangle.

sin(π/4)=√(,2)/2

4. Substitute the value back into the expression for the sine of the negative angle.

sin(−π/4)=−√(,2)/2

5. Calculate the reciprocal to find the cosecant value.

csc(−π/4)=1/(−√(,2)/2)

6. Simplify the fraction by multiplying by the reciprocal of the denominator and rationalizing if necessary.

csc(−π/4)=−2/√(,2)

csc(−π/4)=−√(,2)

Final Answer

csc(−π/4)=−√(,2)

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