Evaluate csc(60 degrees )

Problem

csc(60^∘)

Solution

1. Identify the definition of the cosecant function in terms of the sine function.

csc(θ)=1/sin(θ)

2. Substitute the given angle of 60^∘ into the identity.

csc(60^∘)=1/sin(60^∘)

3. Recall the exact value of sin(60^∘) from the unit circle or a special right triangle (30^∘−60^∘−90^∘.

sin(60^∘)=√(,3)/2

4. Calculate the reciprocal of the sine value to find the cosecant.

csc(60^∘)=1/√(,3)/2

csc(60^∘)=2/√(,3)

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

csc(60^∘)=(2⋅√(,3))/(√(,3)⋅√(,3))

csc(60^∘)=(2√(,3))/3

Final Answer

csc(60^∘)=(2√(,3))/3

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