Use differentials (or equivalently, a linear approximation) to approximate sin(41∘)

Question

Use differentials (or equivalently, a linear approximation) to approximate sin(41∘) as follows: Let 𝑓(𝑥)=sin(𝑥) and find the equation of the tangent line to 𝑓(𝑥) at a "nice" point near 41∘. Then use this to approximate sin(41∘).

Answer 1

41° = 41π/180

We approximate using known values at 45°. sin(45°) = cos(45°) = √2/2 45°= 45π/180 = π/4

(sin x)'= cos x

Then, the approximation of sin(41°) = sin(45°) + (41π/180 - π/4) cos(45°) = √2/2 + (-4π/180)√2/2 =

√2/2 - √2 π/90

The actual value is about 0.6561, while the approximation is approximately .6577.