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

Want more problems? Check here!