Find the Exact Value sin(27)

Problem

sin(27^∘)

Solution

1. Identify the angle as 27^∘ which can be expressed as the difference between two angles with known trigonometric values: 45^∘−18^∘

2. Apply the sine subtraction formula sin(A−B)=sin(A)*cos(B)−cos(A)*sin(B)

3. Substitute the values A=45^∘ and B=18^∘ into the formula.

sin(27^∘)=sin(45^∘)*cos(18^∘)−cos(45^∘)*sin(18^∘)

4. Recall the exact values for 45^∘ sin(45^∘)=√(,2)/2 and cos(45^∘)=√(,2)/2

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

6. Substitute these values into the expression.

sin(27^∘)=√(,2)/2⋅√(,10+2√(,5))/4−√(,2)/2⋅(√(,5)−1)/4

7. Simplify the expression by combining the terms over a common denominator of 8

sin(27^∘)=(√(,20+4√(,5))−(√(,10)−√(,2)))/8

8. Factor the first term √(,20+4√(,5)) as √(,4*(5+√(,5)))=2√(,5+√(,5))

sin(27^∘)=(2√(,5+√(,5))−√(,10)+√(,2))/8

Final Answer

sin(27^∘)=(2√(,5+√(,5))−√(,10)+√(,2))/8

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