Find the Exact Value csc(-pi/3)

Problem

csc(−π/3)

Solution

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

csc(θ)=1/sin(θ)

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

sin(−π/3)=−sin(π/3)

3. Determine the exact value of sin(π/3) using the unit circle or special right triangles.

sin(π/3)=√(,3)/2

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

sin(−π/3)=−√(,3)/2

5. Calculate the reciprocal to find the cosecant value.

csc(−π/3)=1/(−√(,3)/2)

6. Simplify the fraction by multiplying by the reciprocal of the denominator.

csc(−π/3)=−2/√(,3)

7. Rationalize the denominator by multiplying the numerator and denominator by √(,3)

csc(−π/3)=−(2√(,3))/3

Final Answer

csc(−π/3)=−(2√(,3))/3

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