Find the Exact Value csc(33)

Problem

csc(33)

Solution

1. Identify the expression as the cosecant of an angle in degrees, which is the reciprocal of the sine function.

csc(33)=1/sin(33)

2. Recognize that 33 is not a standard reference angle (like 30 45 or 60.

3. Determine if the value can be expressed using radicals. Using the angle addition formula 33=18+15 or 33=60−27 the exact value involves nested radicals.

4. Apply the exact value for sin(33) Using the identity sin(33)=sin(18+15) we use sin(18)=(√(,5)−1)/4 and sin(15)=(√(,6)−√(,2))/4

5. Simplify the resulting expression. The exact value of sin(33) is:

sin(33)=((√(,5)−1)*(√(,6)+√(,2))+(10+2√(,5))(1/2)*(√(,6)−√(,2)))/16

6. Invert the sine value to find the cosecant.

csc(33)=16/((√(,5)−1)*(√(,6)+√(,2))+√(,10+2√(,5))*(√(,6)−√(,2)))

Final Answer

csc(33)=16/((√(,5)−1)*(√(,6)+√(,2))+√(,2*(5+√(,5)))*(√(,6)−√(,2)))

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