Question 1.3.2: Linearization of the Sine Function We will see in Chapter 2 ...
Linearization of the Sine Function
We will see in Chapter 2 that the models of many mechanical systems involve the sine function sin θ, which is nonlinear. Obtain three linear approximations of f (θ) = sin θ, one valid near θ = 0, one near θ = π/3 rad (60°), and one near θ = 2π/3 rad (120°).
The essence of the linearization technique is to replace the plot of the nonlinear function with a straight line that passes through the reference point and has the same slope as the nonlinear function at that point. Figure 1.3.3 shows the sine function and the three straight lines obtained with this technique. Note that the slope of the sine function is its derivative, d sin θ/dθ = cos θ, and thus the slope is not constant but varies with θ.
Consider the first reference point, θ = 0. At this point the sine function has the value sin 0 = 0, the slope is cos 0 = 1, and thus the straight line passing through this point with a slope of 1 is f (θ) = θ. This is the linear approximation of f (θ) = sin θ valid near θ = 0, line A in Figure 1.3.3. Thus we have derived the commonly seen small-angle approximation sin θ ≈ θ.
Next consider the second reference point, θ = π/3 rad. At this point the sine function has the value sin π/3 = 0.866, the slope is cos π/3 = 0.5, and thus the straight line passing through this point with a slope of 0.5 is f (θ) = 0.5(θ − π/3) + 0.866, line B in Figure 1.3.3. This is the linear approximation of f (θ) = sin θ valid near θ = π/3.
Now consider the third reference point, θ = 2π/3 rad. At this point the sine function has the value sin 2π/3 = 0.866, the slope is cos 2π/3 = −0.5, and thus the straight line passing through this point with a slope of −0.5 is f (θ) = −0.5(θ − 2π/3) + 0.866, line C in Figure 1.3.3. This is the linear approximation of f (θ) = sin θ valid near θ = 2π/3.
