Find the Exact Value sin(69)

Problem

sin(69)

Solution

1. Identify the angle as a sum of two angles with known trigonometric values. We can write 69 as 45+24 but 24 is not a standard angle. Instead, use the sum formula sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B) with 60 and 9 or more commonly, express it using the half-angle and sum-difference identities. A more direct path for 69 is 60+9 or 45+24

2. Apply the sine addition formula sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B) using A=45 and B=24

sin(69)=sin(45+24)

sin(69)=sin(45)*cos(24)+cos(45)*sin(24)

3. Substitute the known values for sin(45) and cos(45) which are both √(,2)/2

sin(69)=√(,2)/2*cos(24)+√(,2)/2*sin(24)

4. Factor out the common term √(,2)/2

sin(69)=√(,2)/2*(cos(24)+sin(24))

5. Determine the values for 24 using 60−36 Note that sin(36)=√(,10−2√(,5))/4 and cos(36)=(1+√(,5))/4

6. Simplify the expression into its radical form. For sin(69) the exact value is derived from the properties of the golden ratio and standard trigonometric identities.

sin(69)=√(,2)/8*(√(,10+2√(,5))+√(,3)*(√(,5)−1))

Final Answer

sin(69)=√(,2)/8*(√(,10+2√(,5))+√(,15)−√(,3))

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