I understand how to solve this problem ,however cannot figure out why the .5 degrees is being converted to radians:

PROBLEM:

How would one determine the percent uncertainty in theta and in sin of theta for the following:

1) theta = 15.0 degrees +/- 0.5 degrees

SOLUTION:

f(θ) = sin(θ)
f'(θ) = cos(θ)
For θ = 15°
d(cos(θ))
= ±cos(θ) dθ
= ±cos(15°) (0.5°)
= ±0.9659*(0.5*π/180) <---- Why is .5 degrees converted to radians?
= ±0.00843

The reason why the .5 degrees is being converted to radians is because trigonometric functions such as sin and cos operate with radians as their argument, not degrees. In order to use the values in the appropriate units, it is necessary to convert the angle from degrees to radians.

To convert degrees to radians, you can use the formula:

radians = degrees * π / 180

In this case, the uncertainty of ± 0.5 degrees is being converted to radians to ensure consistency and accuracy when calculating the derivative of cos(θ) and determining the percent uncertainty in sin(θ).

The reason the 0.5 degrees is being converted to radians in this case is because the derivative of cosine is in terms of radians, not degrees.

When we use calculus to calculate the uncertainty in a function, we usually express the derivative in terms of the independent variable. In this case, the independent variable is measured in degrees, but the derivative of cosine is defined in terms of radians.

To convert degrees to radians, we use the conversion factor π/180, which tells us that there are π/180 radians in one degree. So, when we convert 0.5 degrees to radians, we multiply it by π/180:

0.5 degrees * (π/180 radians/degree) = 0.5 * (π/180) radians = 0.00873 radians

In the solution, they approximate π/180 as 0.00873. So, when they calculate the uncertainty in cosine at θ = 15°, they multiply the converted uncertainty (0.00873 radians) with the cosine of 15°.