Using differentials, find the approximate values of the following:
cos 61°, it being given that sin 60° = 0.86603 and 1° = 0.01745 radian
cos 61°, it being given that sin 60° = 0.86603 and 1° = 0.01745 radian
Let us assume that f(x) = cos x
Also, let x = 60° so that x + Δx = 61°
⇒ 60° + Δx = 61°
∴ Δx = 1° = 0.01745 radian
On differentiating f(x) with respect to x, we get
We know
When x = 60°, we have
Recall that if y = f(x) and Δx is a small increment in x, then the corresponding increment in y, Δy = f(x + Δx) – f(x), is approximately given as
Here, and Δx = 0.01745
⇒ Δf = (–0.86603)(0.01745)
∴ Δf = –0.0151122
Now, we have f(61°) = f(60°) + Δf
⇒ f(61°) = cos(60°) – 0.0151122
⇒ f(61°) = 0.5 – 0.0151122
∴ f(61°) = 0.4848878
Thus, cos 61° ≈ 0.4848878