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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