Find the Exact Value csc(960)

Problem

csc(960^∘)

Solution

1. Find the coterminal angle by subtracting multiples of 360^∘ until the angle is between 0^∘ and 360^∘

960^∘−2*(360^∘)=960^∘−720^∘=240^∘

2. Determine the quadrant of the angle 240^∘ Since 180^∘<240^∘<270^∘ the angle lies in Quadrant III.

3. Calculate the reference angle for an angle in Quadrant III using the formula (θ_ref)=θ−180^∘

240^∘−180^∘=60^∘

4. Apply the reciprocal identity for cosecant, which is csc(θ)=1/sin(θ)

csc(240^∘)=1/sin(240^∘)

5. Determine the sign of the sine function in Quadrant III. Sine is negative in Quadrant III.

sin(240^∘)=−sin(60^∘)

6. Substitute the known value for sin(60^∘)=√(,3)/2

sin(240^∘)=−√(,3)/2

7. Calculate the reciprocal to find the cosecant value.

csc(240^∘)=1/(−√(,3)/2)=−2/√(,3)

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

−2/√(,3)⋅√(,3)/√(,3)=−(2√(,3))/3

Final Answer

csc(960^∘)=−(2√(,3))/3

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