Kayla claims that she can find an approximate value for sin 1o without using a machine to do any

computations. Explain or show how she can do this (Hint: if you choose to explain the process, please be very specific. If you choose to show how it is done, you can use your calculator to check your approximation).

using:

f(x)≈f(xo)+f′(xo)(x−xo) <-- should be in your text

1° = 1° - 0°
so let xo = 0, then x = 1

we have :
f(x) = sinx
f'(x) = cosx
since we are using Calculus on trig, our x must be in radians
1° = π/180
f(1°) = sin0 + cos(0)(π/180-0)
= 0 + (1)(π/180)
= π/180
= mmmhh, without a "machine for calculation" ??

anyway, π/180 = .017453292
and sin 1° = .017452406, not bad if I may say so!!!!
(error ≈ .000000886)

To find an approximate value for sin 1o without using any computational devices, Kayla can use the concept of small-angle approximation.

The small-angle approximation states that for small angles, the sine of the angle can be approximated to the value of the angle itself, when measured in radians. This approximation holds true since the sine function approaches the value of the angle as the angle approaches zero.

To apply this approximation to find an approximation for sin 1o, Kayla needs to convert the angle to radians.

Since 1o is equivalent to 1/180th of a full circle, and there are 2π radians in a full circle, Kayla can determine the value of 1o in radians as follows:

1o * (2π/360) = 0.0174533 radians (approx.).

Therefore, using the small-angle approximation, sin 1o can be approximated as 0.0174533.

To verify this result, Kayla can use a calculator to find the exact value of sin 1o. On most scientific calculators, the sin function operates in radians mode. By inputting sin(1o) into the calculator, Kayla should obtain a value very close to the small-angle approximation of 0.0174533.