Find the Exact Value sin(78)

Problem

sin(78^∘)

Solution

1. Identify the angle as a sum of two known angles from the unit circle. We can write 78^∘ as 18^∘+60^∘ or 45^∘+33^∘ However, using the sum formula with 30^∘+48^∘ or 45^∘+33^∘ is difficult. Instead, we use the identity sin(78^∘)=cos(12^∘)

2. Apply the half-angle or sum/difference identities. A more direct way to find the exact value of sin(78^∘) is to use the values for sin(18^∘) and cos(18^∘) combined with 60^∘

3. Recall the exact value of sin(18^∘) which is derived from a golden triangle:

sin(18^∘)=(√(,5)−1)/4

4. Calculate cos(18^∘) using the identity cos^2(θ)+sin^2(θ)=1

cos(18^∘)=√(,1−((√(,5)−1)/4)^2)=√(,10+2√(,5))/4

5. Use the sine addition formula sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B) with A=60^∘ and B=18^∘

sin(78^∘)=sin(60^∘+18^∘)

6. Substitute the known values sin(60^∘)=√(,3)/2 and cos(60^∘)=1/2

sin(78^∘)=√(,3)/2⋅√(,10+2√(,5))/4+1/2⋅(√(,5)−1)/4

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

sin(78^∘)=(√(,30+6√(,5))+√(,5)−1)/8

Final Answer

sin(78^∘)=(√(,30+6√(,5))+√(,5)−1)/8

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