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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