Find the Exact Value sin(33)

Problem

sin(33)

Solution

1. Identify the task as finding the exact trigonometric value of sin(33)

2. Recognize that 33 is not a standard angle (like 30 45 or 60 nor is it a simple multiple or sum of standard angles that results in a basic radical expression.

3. Apply the sum and difference formulas or half-angle formulas if possible. Note that 33=18+15

4. Substitute known exact values for sin(18) and trigonometric functions of 15

5. Recall the exact value sin(18)=(√(,5)−1)/4 and cos(18)=√(,10+2√(,5))/4

6. Recall the exact values sin(15)=(√(,6)−√(,2))/4 and cos(15)=(√(,6)+√(,2))/4

7. Use the sine addition formula sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

8. Calculate the expression:

sin(33)=sin(18)*cos(15)+cos(18)*sin(15)

9. Simplify the resulting radical expression:

sin(33)=(√(,5)−1)/4⋅(√(,6)+√(,2))/4+√(,10+2√(,5))/4⋅(√(,6)−√(,2))/4

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

Final Answer

sin(33)=(√(,30)+√(,10)−√(,6)−√(,2)+(√(,6)−√(,2))√(,10+2√(,5)))/16

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