Mathematics for the liberal arts (2013)
Part I. MATHEMATICS IN HISTORY
Chapter 4. CALCULUS
4.13 Periodic Functions and Trigonometry
Think about riding a Ferris wheel. What are the aspects of the ride that make it enjoyable? As you turn about the wheel, you initially rise up the back side, and the lift of the wheel presses you down into the seat. There’s the moment where you “hang” at the top, and then you fall down the front side.
The sensations of a Ferris wheel come predominantly from the interaction of gravity with the centrepital (due to rotation) force you experience, but you might wonder related questions like “how fast am I falling (vertically) when I go around the wheel?” When is the vertical acceleration the greatest? When is it the least?
The Circular Functions
You may recall from trigonometry that, on a circle of radius r, a point (x, y) is completely determined by the angle θ of a ray from the origin, as in Figure 4.26.
Figure 4.26 The circular functions y = r sin θ and x = r cos θ.
The ratio of y to r is given by the sine function,
Another way to say this is that if you know θ and r, then the y coordinate must be y = r sin θ. Similarly, the cosine function relates the x coordinate to the radius:
Equivalently, x = r cos θ.
Derivatives of Sine and Cosine
If we are to use sine and cosine with our calculus, we’re going to need to know the derivative of each. Although these differentiation formulas can be proven, here we will guess the formulas from graphs.
Note, in calculus we almost always take angles to be measured in radians (that is, a complete circle is 2π radians rather than 360°). Just as you measure temperature in Celsius (or when things really matter, in Kelvin) when doing chemistry, if you want your mathematics to turn out right in calculus class, your best bet is to work in radians.
Look carefully at the sine function in the top graph of Figure 4.27. A small piece of tangent line has been added at each of the multiples of . Notice at , for example, that the slope of the tangent is 0. At the origin, the tangent line appears to have slope 1. Now compare with the points added to the bottom (cosine) graph. The slope of each tangent to sine has been plotted on the cosine graph.
When the slope of sine is zero, the cosine graph has value zero. When the slope of sine is 1, the cosine has value 1. In each case, the slope of the sine curve is precisely the value on the cosine curve. Thus, the cosine function tells the slope of the sine function. That means that the derivative of y = sin t is the function y = cos t.
Figure 4.27 Graphs of y = sin t and y = cos t.
The sine rule: The derivative of f(x) = sin x is f′(x) = cos x.
EXAMPLE 4.38
What is the tangent line to y = sin x when x = π/6?
Solution: For a tangent line, we need a point and a slope. The y coordinate of the point of tangency is y = sin(π/6) = 1/2. The slope is given by the derivative, m = cos(π/6) = . Using the point-slope form of a line, the tangent is
In slope-intercept form, this is
There are several ways to determine the derivative of cosine, and perhaps the most obvious is to do what we did for sine, draw a graph and guess. But we can also derive the rule for cosine from the existing sine rule, because sine and cosine are related by a cofunction identity. For any angle θ, the complementary angle is 90° – θ, and cos θ = sin(90° – θ). Figure 4.28 illustrates this relationship on a triangle.
Figure 4.28 Complementary angles: cos θ = sin(90° θ) = x/r.
If we know the complementary angle formula, we can derive a derivative formula for cosine. First we switch to radians, and then we differentiate using the chain rule:
On the right hand side, the outside function is g(x) = sin x and the inside function is f(x) = π/2 – x. By the chain rule,
There is one last trick, and that is to use a cofunction identity again to replace cos(π/2 – x) with sin x,
The cosine rule: The derivative of f(x) = cos x is f′(x) = – sin x.
EXAMPLE 4.39
What is the derivative of y = x2 cos(3x + 1)?
Solution: All of our usual rules (product rule, chain rule, etc.) apply to trig functions. So, beginning with the product rule,
and continuing with the chain rule,
We now have the tools we need to answer questions about Ferris wheels. The Beijing Great Wheel, planned to be the tallest Ferris wheel in the world if completed, will have a wheel of diameter 99 m and reach a total height of 208 m. One revolution of the wheel requires 20 min (1200 s).
A function that models the height of a point on this wheel, with t given in seconds, is
To verify this, check that t = 0 gives a height of 10 m (the lowest the wheel achieves):
After 10 min (600 s), the function reaches its maximum height,
EXAMPLE 4.40
What is the vertical velocity of a point moving around the Great Wheel at time t = 0,150, 300, or 600 s?
Solution: We need to compute the derivative.
This yields vertical velocity values (in units of m/s) of
t |
y′(t) |
0 |
0 |
150 |
0.3665 |
300 |
0.5134 |
600 |
0 |
EXERCISES
4.71 Find the derivative of each function.
a)
b)
c) x(t) = 30cos(2πt)
d) g(t) =
e) h(t) = cos(ln x)
4.72 In Figure 4.29 the top function is y = cos t, and the bottom function is y = sin t. Where each piece of tangent line occurs on the top graph, draw a corresponding point on the bottom graph representing the slope. Use these values to guess the derivative of cosine.
Figure 4.29 Graphs of y = cos t and y = sin t.
4.73 Write and use the quotient rule to find a derivative formula for the tangent function.
4.74 Write and differentiate to find a derivative formula for the secant function.
4.75 Write and differentiate to find a derivative formula for the cosecant function.
4.76 Write and differentiate to find a derivative formula for the cotangent function.
4.77 From trigonometry, we know the identity sin 2t = 2 sin t cos t.
a) Find the derivative of sin 2t.
b) Find the derivative of 2 sin t cos t.
c) What must be true of your answers from parts (a) and (b)?
d) Plot your answers from a and b on the same graph.
4.78 The horizontal position of a point on the Beijing Great Wheel is modeled by the function
a) Find the horizontal position at times t = 0, 150, 300, and 600 s.
b) Find the horizontal velocity at times t = 0,150, 300, and 600 s.