Question

A student measured an angle θ = 41 ± 1°. convert the value to radians. calculate sin(θ) . evaluate the uncertainty in the measured value of the sin θ. note that the uncertainty in θ, δθ is not equal to the uncertainty in sin θ, δsin θ. one simple way of determining the uncertainty in sin θ is by δsin θ = (sin θmax - sin θmin)/2, where θmax = θ + δθ and θmin = θ - δθ. remember, that any uncertainty calculation is just an estimate, so don't report more than two significant digits!

Answer 1

To convert an angle from degrees to radians, multiply by π/180. To calculate sin(θ), use a calculator or trigonometric table. To evaluate the uncertainty in sin(θ), use the formula δsin(θ) = (sin(θ + δθ) - sin(θ - δθ))/2.

Step-by-step explanation:

To convert an angle measured in degrees to radians, you can use the formula:

Radians = (θ * π) / 180

Given that θ = 41° ± 1°, we can calculate the radians as:

Radians = (41° * π) / 180 = 0.715585 radians

To calculate sin(θ), we can use the sin function in a calculator or by using a trigonometric table:

sin(θ) ≈ 0.656059

To evaluate the uncertainty in the measured value of sin(θ), we can use the formula:

Δsin(θ) = (sin(θ + δθ) - sin(θ - δθ)) / 2

Substituting the values:

Δsin(θ) = (sin(42°) - sin(40°)) / 2 ≈ 0.024958

Answer 2

Angle = 41 degrees

Conversion to radians:
41° = 41*π/180 = 0.7156 radians
1° = 1*π/180 = 0.0175
Angle measured in radians = 0.7156+/-0.0175

Sin (angle):
Sin (0.7156) = 0.6561

Uncertainty in Sin (0.7156) = (Sin∅ max - Sin∅ min)/2 = (Sin 0.7330 - Sin 0.6981)/ 2 = 0.0132

Sin ∅ = 0.06561+/-0.0132