Find the Exact Value sin(81)

Problem

sin(81)

Solution

1. Identify the angle as 81 Since this is not a standard reference angle, we can express it as the sum or difference of angles with known trigonometric values, such as 81=45+36

2. Apply the sine addition formula which states sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

3. Substitute A=45 and B=36 into the formula.

sin(81)=sin(45+36)

sin(81)=sin(45)*cos(36)+cos(45)*sin(36)

4. Use known values for the trigonometric functions. We know sin(45)=√(,2)/2 and cos(45)=√(,2)/2 For 36 the exact values are cos(36)=(1+√(,5))/4 and sin(36)=√(,10−2√(,5))/4

5. Substitute these values into the expression.

sin(81)=√(,2)/2⋅(1+√(,5))/4+√(,2)/2⋅√(,10−2√(,5))/4

6. Simplify the expression by combining the terms over a common denominator.

sin(81)=(√(,2)+√(,10))/8+√(,20−4√(,5))/8

sin(81)=(√(,2)+√(,10)+2√(,5−√(,5)))/8

Final Answer

sin(81)=(√(,2)+√(,10)+2√(,5−√(,5)))/8

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