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 cos2(θ)+sin2(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

Want more problems? Check here!