Final answer:
To approximate sin(61°), use sin(60°) as a base, convert angles to radians, and apply differentials with the derivative cos(x). Sin(61°) ≈ (√3/2) + (1/2)×(π/180), giving an exact form approximation.
Step-by-step explanation:
To approximate sin(61°) using differentials, we can use sin(60°) as a starting point since it's close to 61° and has an exact value that we know. The formula to use a differential to approximate a function value is Δy ≈ f'(x)Δx where Δx is the small change in x and f'(x) is the derivative of the function at x.
First, we convert the angle to radians because we must use radians when working with trigonometric functions in calculus. So, 60° = π/3 radians. The derivative of sin(x) is cos(x), so we will need the value of cos(60°).
We then calculate the differential:
- sin'(x) = cos(x)
- sin'(60°) = cos(60°) = 1/2
- Δx = 61° - 60° = 1° = (π/180) radians
- Δy ≈ cos(60°) × (π/180) = (1/2) × (π/180)
Now, since sin(60°) = √3/2, the approximate value of sin(61°) is:
sin(61°) ≈ sin(60°) + Δy ≈ (√3/2) + (1/2)×(π/180)
This gives us the approximation in exact form.
Finally, check if the answer makes sense by considering that the approximation should be slightly higher than sin(60°) because sine increases between 60° and 61°.