Graphs of sine and cosine
In the last chapter, ${\sin t}$ and ${\cos t}$ were the coordinates of a point moving around the unit circle. Now we graph them as functions. As is usual for graphs, we call the input ${x}$ and the output ${y}$. So we graph ${y=\sin x}$ and ${y=\cos x}$, with ${x}$ in radians.
A table of values
Here are some values from the unit circle, rounded to two decimal places:
| ${x}$ | ${\sin x}$ | ${\cos x}$ |
|---|---|---|
| ${0}$ | ${0}$ | ${1}$ |
| ${\dfrac{\pi}{6}}$ | ${0.5}$ | ${0.87}$ |
| ${\dfrac{\pi}{2}}$ | ${1}$ | ${0}$ |
| ${\dfrac{5\pi}{6}}$ | ${0.5}$ | ${-0.87}$ |
| ${\pi}$ | ${0}$ | ${-1}$ |
| ${\dfrac{7\pi}{6}}$ | ${-0.5}$ | ${-0.87}$ |
| ${\dfrac{3\pi}{2}}$ | ${-1}$ | ${0}$ |
| ${\dfrac{11\pi}{6}}$ | ${-0.5}$ | ${0.87}$ |
| ${2\pi}$ | ${0}$ | ${1}$ |
As ${x}$ goes from ${0}$ to ${2\pi}$, the point on the unit circle goes once around. Its height, ${\sin x}$, rises from ${0}$ to ${1}$, falls to ${-1}$, and comes back to ${0}$.
The graph of sine
Plot the points from the table, and join them with a smooth curve. This gives one period of the graph:
Sine has period ${2\pi}$: its values repeat every ${2\pi}$. So the same piece repeats forever to the left and to the right. The graph is a smooth, endless wave:
The graph of cosine
The cosine column of the table gives the cosine graph in the same way. It starts at its high point, ${(0,1)}$:
The two graphs have the same shape. In fact, the cosine graph is the sine graph moved ${\dfrac{\pi}{2}}$ to the left. Each high point of sine, such as ${\left(\dfrac{\pi}{2},1\right)}$, lines up with a high point of cosine, ${(0,1)}$.
Moving a graph ${\dfrac{\pi}{2}}$ to the left means replacing ${x}$ by ${x+\dfrac{\pi}{2}}$. So
${\cos x=\sin\left(x+\dfrac{\pi}{2}\right)}$
A curve with this wave shape is called a sine curve, or a sinusoid, whether it comes from sine or cosine.
Properties of the graphs
The graphs show the facts from the unit circle at a glance:
| ${y=\sin x}$ | ${y=\cos x}$ | |
|---|---|---|
| Domain | all real numbers | all real numbers |
| Range | ${[-1,1]}$ | ${[-1,1]}$ |
| Period | ${2\pi}$ | ${2\pi}$ |
| Zeros | ${0}$, ${\pm\pi}$, ${\pm 2\pi}$, … | ${\pm\dfrac{\pi}{2}}$, ${\pm\dfrac{3\pi}{2}}$, … |
| Symmetry | odd: about the origin | even: about the ${y}$-axis |
- Both graphs stay between the lines ${y=-1}$ and ${y=1}$.
- The sine graph is symmetric about the origin, because sine is odd: ${\sin(-x)=-\sin x}$.
- The cosine graph is a mirror image of itself across the ${y}$-axis, because cosine is even: ${\cos(-x)=\cos x}$.
Key points
To sketch one period quickly, split the interval ${[0,2\pi]}$ into four equal parts. This gives five key points: the zeros, the high point, and the low point.
Sine: ${(0,0)}$, ${\left(\dfrac{\pi}{2},1\right)}$, ${(\pi,0)}$, ${\left(\dfrac{3\pi}{2},-1\right)}$, ${(2\pi,0)}$
Cosine: ${(0,1)}$, ${\left(\dfrac{\pi}{2},0\right)}$, ${(\pi,-1)}$, ${\left(\dfrac{3\pi}{2},0\right)}$, ${(2\pi,1)}$
Plot the five points, and join them with a smooth wave. Remember that a sine curve is rounded at its high and low points, not pointed.
The rules for moving and flipping graphs from Graphs of functions in College Algebra work for these graphs too. The next two examples use them. The next lesson does much more with them.
Example 1: Sketch one period of ${y=-\sin x}$.
Solution:
Each ${y}$-value of ${\sin x}$ changes sign. So the graph of ${y=\sin x}$ is flipped over the ${x}$-axis. The key points become
${(0,0)}$, ${\left(\dfrac{\pi}{2},-1\right)}$, ${(\pi,0)}$, ${\left(\dfrac{3\pi}{2},1\right)}$, ${(2\pi,0)}$
Example 2: Sketch one period of ${y=1+\cos x}$. Find its range.
Solution:
Add ${1}$ to each ${y}$-value of ${\cos x}$. The graph of ${y=\cos x}$ moves up ${1}$. The key points become
${(0,2)}$, ${\left(\dfrac{\pi}{2},1\right)}$, ${(\pi,0)}$, ${\left(\dfrac{3\pi}{2},1\right)}$, ${(2\pi,2)}$
The values of ${\cos x}$ go from ${-1}$ to ${1}$. Adding ${1}$, the values of ${1+\cos x}$ go from ${0}$ to ${2}$. The range is ${[0,2]}$.
Example 3: Use the graph of ${y=\sin x}$ to find all ${x}$ in ${[0,2\pi]}$ with ${\sin x>\dfrac{1}{2}}$.
Solution:
Draw the line ${y=\dfrac{1}{2}}$, and find where the curve is above it. From Chapter 1, ${\sin x=\dfrac{1}{2}}$ at ${x=\dfrac{\pi}{6}}$ (quadrant I) and ${x=\dfrac{5\pi}{6}}$ (quadrant II, reference angle ${\dfrac{\pi}{6}}$).
The curve is above the line between these two points. So ${\sin x>\dfrac{1}{2}}$ when
${\dfrac{\pi}{6}<x<\dfrac{5\pi}{6}}$
The ends are not included, because there ${\sin x}$ equals ${\dfrac{1}{2}}$.
Summary
- The graphs of ${y=\sin x}$ and ${y=\cos x}$ are waves with period ${2\pi}$ and range ${[-1,1]}$.
- Sine starts at ${(0,0)}$ and rises. Cosine starts at its high point ${(0,1)}$.
- The cosine graph is the sine graph moved ${\dfrac{\pi}{2}}$ to the left.
- Sketch one period from five key points, found by splitting ${[0,2\pi]}$ into four equal parts.
- Sine is odd (symmetric about the origin). Cosine is even (symmetric about the ${y}$-axis).