Graphs of Sine and Cosine

The graphs of sine and cosine are smooth waves that repeat every 2\pi radians (360°). The graph of y = \sin x starts at 0, rises to 1 at \frac{\pi}{2}, returns to 0 at \pi, falls to −1 at \frac{3\pi}{2} and is back to 0 at 2\pi. The graph of y = \cos x is the same wave shifted left \frac{\pi}{2}, so it starts at 1.

Before this lesson: You should know The Unit Circle, Parent Functions and Radians.

Unwrapping the Unit Circle

Picture a point moving counterclockwise around the unit circle. Its height is \sin x, where x is the angle in radians. As the point goes around once, the height climbs from 0 to 1, drops to −1 and returns to 0. Plotting the height against the angle gives the sine wave.

The point’s left-right position is \cos x. It starts at 1, moves to −1 halfway around and returns to 1. Plotting that gives the cosine wave. Because the point repeats its path every full turn, both graphs repeat every 2\pi.

Key Points and Features

One period is split into four equal parts by five key points. Plot them and connect them with a smooth curve.

The period is the length of one full cycle. The amplitude is half the distance from the lowest point to the highest point, which is 1 for both parent graphs. The domain and range show the graphs go on forever left and right but stay between −1 and 1.

Worked Examples

Example 1: Sine values between the key points

Make a table of values for y = \sin x from 0 to \pi, using steps of \frac{\pi}{6}, and describe the graph.

\sin 0 = 0

\sin \frac{\pi}{6} = \frac{1}{2} = 0.5

\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \approx 0.87

\sin \frac{\pi}{2} = 1

\sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2} \approx 0.87

\sin \frac{5\pi}{6} = \frac{1}{2} = 0.5

\sin \pi = 0

Answer: The graph rises from 0 to a peak of 1 at \frac{\pi}{2}, then falls back to 0 at \pi. The values are symmetric around \frac{\pi}{2}, so the hump is symmetric too.

Example 2: Key points of cosine over two periods

List the key points of y = \cos x from -2\pi to 2\pi.

(0, 1),\ (\frac{\pi}{2}, 0),\ (\pi, -1),\ (\frac{3\pi}{2}, 0),\ (2\pi, 1)

\cos(-x) = \cos x

(-\frac{\pi}{2}, 0),\ (-\pi, -1),\ (-\frac{3\pi}{2}, 0),\ (-2\pi, 1)

Answer: (-2\pi, 1), (-\frac{3\pi}{2}, 0), (-\pi, -1), (-\frac{\pi}{2}, 0), (0, 1), (\frac{\pi}{2}, 0), (\pi, -1), (\frac{3\pi}{2}, 0), (2\pi, 1)

Example 3: Use the period

Find every x with 0 \le x \le 4\pi where \sin x = 1.

x = \frac{\pi}{2}

\frac{\pi}{2} + 2\pi = \frac{\pi}{2} + \frac{4\pi}{2} = \frac{5\pi}{2}

Adding another period gives \frac{9\pi}{2}, which is more than 4\pi = \frac{8\pi}{2}, so stop.

Answer: x = \frac{\pi}{2} and x = \frac{5\pi}{2}

Example 4: Where the graph rises and falls

On the interval [0, 2\pi], where is y = \cos x increasing, and where is it decreasing?

x = 0

x = \pi

x = 2\pi

Answer: decreasing on (0, \pi) and increasing on (\pi, 2\pi). This uses the ideas from key features of functions.

Example 5: Word problem with a buoy

A buoy bobs on waves. Its height above calm water, in meters, is h(t) = \sin t, where t is time in seconds. During the first 7 seconds, when is the buoy highest, and when is it lowest?

The highest value of sine is 1, at t = \frac{\pi}{2}

\frac{\pi}{2} \approx 1.57

The next peak is at \frac{5\pi}{2} \approx 7.85, which is after 7 seconds.

The lowest value of sine is −1, at t = \frac{3\pi}{2}

\frac{3\pi}{2} \approx 4.71

Answer: The buoy is highest (1 meter above calm water) at about 1.57 seconds and lowest (1 meter below) at about 4.71 seconds.

Common Mistakes

Related Methods

Cosine is a shifted sine

Shift the sine graph left by \frac{\pi}{2} and it lands exactly on the cosine graph: \cos x = \sin (x + \frac{\pi}{2}). This is one reason the two graphs share every feature except their starting point.

Degrees on the x-axis

You can also label the x-axis in degrees: 0°, 90°, 180°, 270° and 360°. The shape is the same; only the labels change. Calculus uses radians, so practice both.

Changing the shape

In the next lessons, numbers in y = a \sin (bx - c) + d stretch and shift the parent graph, just like the transformations of functions from Algebra 2.

Practice Problems

1. Find \sin \frac{\pi}{2}.
2. Find \cos \pi.
3. List the zeros of y = \sin x on [0, 2\pi].
4. List the zeros of y = \cos x on [0, 2\pi].
5. Where on [0, 2\pi] does y = \cos x reach its maximum?
6. What is the range of y = \sin x?
7. Find \sin(-\frac{\pi}{2}) and explain what it shows about the sine graph.
8. Find \cos(-\frac{\pi}{3}).
9. Find every x on [0, 2\pi] where \cos x = -1.
10. A mass on a spring moves up and down. Its position is y = \cos t centimeters at time t seconds. Find the first time after 0 when the mass is at y = 0, and the first time it is at its lowest point.

\cos(-\frac{\pi}{3}) = \cos \frac{\pi}{3}

\cos \frac{\pi}{3} = \frac{1}{2}

FAQ

What do the graphs of sine and cosine look like?

Both are smooth waves that go up and down between −1 and 1 forever. Sine starts at 0 and rises; cosine starts at its peak of 1 and falls.

What is the period of sine and cosine?

The period of both y = \sin x and y = \cos x is 2\pi radians, or 360°. After that, the graphs repeat.

What is the difference between the sine and cosine graphs?

They have the same shape, but cosine is shifted \frac{\pi}{2} to the left. So sine passes through (0, 0) and cosine passes through (0, 1).

What are the 5 key points of a sine graph?

For one period of y = \sin x, they are (0, 0), (\frac{\pi}{2}, 1), (\pi, 0), (\frac{3\pi}{2}, -1) and (2\pi, 0).

Summary

• y = \sin x and y = \cos x are waves with period 2\pi and range [-1, 1].
• Sine key points: 0, 1, 0, −1, 0 at 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi.
• Cosine key points: 1, 0, −1, 0, 1 at the same x-values.
• Sine is odd (symmetric about the origin); cosine is even (symmetric about the y-axis).
• Add or subtract 2\pi to find repeats of any point.

Previous lesson: The Unit Circle

Next lesson: Amplitude and Period

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