Approximate sin56

Question

Use differentials (or equivalently, a linear approximation) to approximate sin(56∘) as follows: Let 𝑓(𝑥)=sin(𝑥) and find the equation of the tangent line to 𝑓(𝑥) at a "nice" point near 56∘. Then use this to approximate sin(56∘).

Answer 1

We know that sin60º = √3/2 and 60° = π/3 radians.

56º - 60º = -4º = -4(π/180) = -π/25

56° = 56(π/180) = 14π/45

Let f(x) = sinx

sin56º = f(14π/45) ≈ L(π/3) = f(π/3) + f'(π/3)(-π/25) = sin(π/3) + cos(π/3)(-π/25)

= √3/2 + (1/2)(-π/25)