2.2: The Sine and Cosine Function

The sine and cosine functions

The sine and cosine functions are among the most important functions in all of mathematics. Sometimes called the circular functions due to their definition on the unit circle, these periodic functions play a key role in modeling repeating phenomena such as tidal elevations, the behavior of an oscillating mass attached to a spring, or the location of a point on a bicycle tire. Like polynomial and exponential functions, the sine and cosine functions are considered basic functions, ones that are often used in building more complicated functions. As such, we would like to know formulas for $\frac{d}{dx} [\sin(x)]$ and $\frac{d}{dx} [\cos(x)]\text{,}$ and the next two activities lead us to that end.

Activity $2.2.2$

Consider the function $f(x) = \sin(x)\text{,}$ which is graphed in Figure $2.2.2$ below. Note carefully that the grid in the diagram does not have boxes that are $1 \times 1\text{,}$ but rather approximately $1.57 \times 1\text{,}$ as the horizontal scale of the grid is $\pi/2$ units per box.

1. At each of $x = -2\pi, -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\text{,}$ use a straightedge to sketch an accurate tangent line to $y = f(x)\text{.}$
2. Use the provided grid to estimate the slope of the tangent line you drew at each point. Pay careful attention to the scale of the grid.
3. Use the limit definition of the derivative to estimate $f'(0)$ by using small values of $h\text{,}$ and compare the result to your visual estimate for the slope of the tangent line to $y = f(x)$ at $x = 0$ in (b). Using periodicity, what does this result suggest about $f'(2\pi)\text{?}$ about $f'(-2\pi)\text{?}$
4. Based on your work in (a), (b), and (c), sketch an accurate graph of $y = f'(x)$ on the axes adjacent to the graph of $y = f(x)\text{.}$
5. What familiar function do you think is the derivative of $f(x) = \sin(x)\text{?}$

Activity $2.2.3$

Consider the function $g(x) = \cos(x)\text{,}$ which is graphed in Figure $2.2.3$ below. Note carefully that the grid in the diagram does not have boxes that are $1 \times 1\text{,}$ but rather approximately $1.57 \times 1\text{,}$ as the horizontal scale of the grid is $\pi/2$ units per box.

1. At each of $x = -2\pi, -\frac{3\pi}{2}, -\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\text{,}$ use a straightedge to sketch an accurate tangent line to $y = g(x)\text{.}$
2. Use the provided grid to estimate the slope of the tangent line you drew at each point. Again, note the scale of the axes and grid.
3. Use the limit definition of the derivative to estimate $g'(\frac{\pi}{2})$ by using small values of $h\text{,}$ and compare the result to your visual estimate for the slope of the tangent line to $y = g(x)$ at $x = \frac{\pi}{2}$ in (b). Using periodicity, what does this result suggest about $g'(-\frac{3\pi}{2})\text{?}$ can symmetry on the graph help you estimate other slopes easily?
4. Based on your work in (a), (b), and (c), sketch an accurate graph of $y = g'(x)$ on the axes adjacent to the graph of $y = g(x)\text{.}$
5. What familiar function do you think is the derivative of $g(x) = \cos(x)\text{?}$

The results of the two preceding activities suggest that the sine and cosine functions not only have beautiful connections such as the identities $\sin^2(x) + \cos^2(x) = 1$ and $\cos(x - \frac{\pi}{2}) = \sin(x)\text{,}$ but that they are even further linked through calculus, as the derivative of each involves the other. The following rules summarize the results of the activities¹.

These two rules may be formally proved using the limit definition of the derivative and the expansion identities for $\sin(x+h)$ and $\cos(x+h)\text{.}$

Sine and Cosine Functions

For all real numbers $x\text{,}$

$$\frac{d}{dx} [\sin(x)] = \cos(x) \ \ \text{and} \ \ \frac{d}{dx} [\cos(x)] = -\sin(x)\text{.}$$

We have now added the sine and cosine functions to our library of basic functions whose derivatives we know. The constant multiple and sum rules still hold, of course, as well as all of the inherent meaning of the derivative.

Activity $2.2.4$

Answer each of the following questions. Where a derivative is requested, be sure to label the derivative function with its name using proper notation.

1. Determine the derivative of $h(t) = 3\cos(t) - 4\sin(t)\text{.}$
2. Find the exact slope of the tangent line to $y = f(x) = 2x + \frac{\sin(x)}{2}$ at the point where $x = \frac{\pi}{6}\text{.}$
3. Find the equation of the tangent line to $y = g(x) = x^2 + 2\cos(x)$ at the point where $x = \frac{\pi}{2}\text{.}$
4. Determine the derivative of $p(z) = z^4 + 4^z + 4\cos(z) - \sin(\frac{\pi}{2})\text{.}$
5. The function $P(t) = 24 + 8\sin(t)$ represents a population of a particular kind of animal that lives on a small island, where $P$ is measured in hundreds and $t$ is measured in decades since January 1, 2010. What is the instantaneous rate of change of $P$ on January 1, 2030? What are the units of this quantity? Write a sentence in everyday language that explains how the population is behaving at this point in time.

Summary

• For an exponential function $f(x) = a^x$ $(a \gt 1)\text{,}$ the graph of $f'(x)$ appears to be a scaled version of the original function. In particular, careful analysis of the graph of$f(x) = 2^x\text{,}$ suggests that $\frac{d}{dx}[2^x] = 2^x \ln(2)\text{,}$ which is a special case of the rule we stated in Section 2.1.
• By carefully analyzing the graphs of $y = \sin(x)$ and $y = \cos(x)\text{,}$ and by using the limit definition of the derivative at select points, we found that $\frac{d}{dx} [\sin(x)] = \cos(x)$ and $\frac{d}{dx} [\cos(x)] = -\sin(x)\text{.}$
• We note that all previously encountered derivative rules still hold, but now may also be applied to functions involving the sine and cosine. All of the established meaning of the derivative applies to these trigonometric functions as well.