Find the Exact Value csc(-300)

Problem

csc(−300)

Solution

1. Use the odd function property of the cosecant function, which states csc(−θ)=−csc(θ)

csc(−300)=−csc(300)

2. Find the reference angle for 300 Since 300 is in the fourth quadrant, the reference angle is 360−300=60

Reference Angle=60

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

csc(300)=−csc(60)

4. Substitute this back into the expression from step 1.

−csc(300)=−(−csc(60))

csc(−300)=csc(60)

5. Evaluate the cosecant of 60 using the reciprocal identity csc(θ)=1/sin(θ)

sin(60)=√(,3)/2

csc(60)=2/√(,3)

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

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

Final Answer

csc(−300)=(2√(,3))/3

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