Simplify csc(240)

Problem

csc(240)

Solution

1. Identify the reference angle by finding the distance from the angle to the nearest x-axis. Since 240 is in the third quadrant, the reference angle is 240−180=60

2. Determine the sign of the cosecant function in the third quadrant. In the third quadrant, sine is negative, and since csc(θ)=1/sin(θ) the cosecant is also negative.

3. Apply the reference angle and sign to rewrite the expression.

csc(240)=−csc(60)

4. Evaluate the cosecant of the reference angle using the reciprocal of the sine function.

sin(60)=√(,3)/2

csc(60)=2/√(,3)

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

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

6. Combine the negative sign with the evaluated value.

csc(240)=−(2√(,3))/3

Final Answer

csc(240)=−(2√(,3))/3

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