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
2) theta = 75.0 degrees +/- 0.5 degrees
I'm not sure whether I divide or use some kind of formula.
I don't see "percentage uncertainty" in this answer. It looks to be just the uncertainty.
To determine the percent uncertainty in theta and in sin of theta, you can use the following formula:
Percent uncertainty = (uncertainty / measurement) x 100%
For example:
1) For theta = 15.0 degrees +/- 0.5 degrees:
Percent uncertainty in theta = (0.5 / 15.0) x 100% = 3.33%
Since the sine function is dimensionless, the uncertainty in theta (in degrees) is directly applicable to sin of theta.
Percent uncertainty in sin(theta) = 3.33%
2) For theta = 75.0 degrees +/- 0.5 degrees:
Percent uncertainty in theta = (0.5 / 75.0) x 100% = 0.67%
Similarly, the percent uncertainty in sin(theta) will be the same as the percent uncertainty in theta.
Percent uncertainty in sin(theta) = 0.67%
So, for both cases, the percent uncertainty in theta and in sin(theta) will be equal to the percent uncertainty in the measurement of theta.
To determine the percent uncertainty in theta and in sin(theta), we need to apply the formula for percent uncertainty:
Percent Uncertainty = (Uncertainty / Measurement) * 100
Let's calculate the percent uncertainty for each case:
1) For theta = 15.0 degrees +/- 0.5 degrees:
Percent Uncertainty in theta = (0.5 / 15.0) * 100
= 0.0333 * 100
= 3.33%
To find the percent uncertainty in sin(theta), we need to calculate sin(15.0 degrees) first.
sin(theta) = sin(15.0 degrees)
≈ 0.2588
Percent Uncertainty in sin(theta) = (Uncertainty / Measurement) * 100
= (0.5 / 0.2588) * 100
≈ 193.15%
2) For theta = 75.0 degrees +/- 0.5 degrees:
Percent Uncertainty in theta = (0.5 / 75.0) * 100
= 0.0067 * 100
= 0.67%
To find the percent uncertainty in sin(theta), we need to calculate sin(75.0 degrees) first.
sin(theta) = sin(75.0 degrees)
≈ 0.9659
Percent Uncertainty in sin(theta) = (Uncertainty / Measurement) * 100
= (0.5 / 0.9659) * 100
≈ 51.76%
So, the percent uncertainty in theta for the first case is approximately 3.33%, and for the second case, it is approximately 0.67%. And the percent uncertainty in sin(theta) for the first case is approximately 193.15%, and for the second case, it is approximately 51.76%.
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)
= ±0.00843
Problem #2 can be calculated similarly.