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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