Graphing sine and cosine

When you reach for it

On the unit circle a sine is found one angle at a time: choose the angle, find its terminal point, read the y-coordinate (Trig functions on the unit circle). Many questions are about all angles at once. Where is the sine positive? Where is it growing? Which angles between 0 and 20 put the terminal point at the very top of the circle? Which angles have \displaystyle \sin x = \frac{1}{2}? A single point answers none of these quickly. A graph answers them all by looking, the way the graph of any function shows its whole behavior at once.

Measured in radians, an angle is a real number (Sine, cosine and tangent as functions of a real number), so sine and cosine are functions from real numbers to real numbers and can be drawn on ordinary axes. This lesson draws y = \sin x and y = \cos x themselves, with no stretching or shifting. It runs over two days. Part 1 (Examples 1 to 4) builds the sine graph from the circle. Part 2 (Examples 5 to 7) builds the cosine graph and the five key points that pin down one cycle of either.

One point about letters. On these graphs x is the input, an angle in radians, exactly as for any function graphed in the plane, and y is the output. The x on the axis is not the x-coordinate of the terminal point.

The procedure

Part 1: unrolling the circle into the sine graph

Let the angle x grow from 0 to 2\pi. Its terminal point starts at (1, 0) and travels once counterclockwise around the unit circle. At every moment the point has a height, its y-coordinate, which is \sin x. Plot each pair (x, \sin x): the angle across, the height up. That is the whole idea. The graph is a record of the height of a point going around a circle, with the circle’s turn laid out flat along the horizontal axis.

The special angles give exact heights to plot:

Here \displaystyle \frac{\sqrt{2}}{2} \approx 0.71 and \displaystyle \frac{\sqrt{3}}{2} \approx 0.87. Read quadrant by quadrant:

• From 0 to \displaystyle \frac{\pi}{2} the point climbs from (1, 0) to (0, 1), so the graph rises from height 0 to height 1.
• From \displaystyle \frac{\pi}{2} to \pi it comes down the left side to (-1, 0), so the graph falls back to 0. The terminal points of x and \pi - x are mirror images across the y-axis (Unit circle symmetry and even-odd identities), so they have the same height, and this piece of the graph is the mirror image of the first piece across the line \displaystyle x = \frac{\pi}{2}.
• From \pi to 2\pi the point is below the x-axis, falling to (0, -1) at \displaystyle \frac{3\pi}{2} and rising back to (1, 0). The graph dips below the axis to -1 and returns to 0. Example 1 fills in these heights.

Why the curve is rounded. The heights do not change at a steady rate. Over the first sixth of a half turn, from 0 to \displaystyle \frac{\pi}{6}, the average rate of change of the height is

Over an input interval of the same width just below the top, from \displaystyle \frac{\pi}{3} to \displaystyle \frac{\pi}{2}, it is

The same amount of turning gains almost four times as much height near the start. The reason is the direction the point is moving. At (1, 0) it moves straight up, so all of its motion goes into height. Near (0, 1) it moves almost sideways, so its height barely changes. The graph leaves the origin climbing steeply and flattens into a rounded top at \displaystyle \left(\frac{\pi}{2}, 1\right). It is not made of straight pieces, and it has no corners.

Beyond one turn. Adding 2\pi to an angle adds one full turn and lands on the same terminal point (Periodicity and the periodicity identities), so \sin(x + 2\pi) = \sin x. The piece of graph over [0, 2\pi] is therefore copied over [2\pi, 4\pi], over [-2\pi, 0], and so on forever in both directions. Negative x means clockwise turning. One repetition is a cycle, and its width 2\pi is the period (Periodic functions).

Facts about y = \sin x, each read from the circle:

• Domain all real numbers, since every angle has a terminal point.
• Range [-1, 1], since heights on the unit circle run from -1 to 1. Each value strictly between -1 and 1 is taken twice in every cycle; 1 and -1 are taken once each.
• Zeros at x = n\pi for every integer n, where the point is on the x-axis.
• Highest points (x, 1) at \displaystyle x = \frac{\pi}{2} + 2\pi n; lowest points (x, -1) at \displaystyle x = \frac{3\pi}{2} + 2\pi n, for integers n.
• Period 2\pi.
• Symmetric about the origin. The angle -x is x reflected across the x-axis, so its height is the opposite: \sin(-x) = -\sin x. Each point (x, y) of the graph comes with (-x, -y), so sine is an odd function (Even and odd functions).

Part 2: cosine, and the five key points

The cosine graph comes from the same moving point, recording its x-coordinate instead of its height. The point starts at (1, 0), so the graph starts at 1. As the point moves up and to the left through Quadrant I its x-coordinate falls to 0 at \displaystyle \frac{\pi}{2}; it keeps falling to -1 at \pi, then climbs back through 0 at \displaystyle \frac{3\pi}{2} to 1 at 2\pi.

The same shape, a quarter turn ahead. Turning a point a quarter turn counterclockwise about the origin sends (a, b) to (-b, a): the new point is just as far from the origin and at right angles to the old one, as (1, 0) \to (0, 1) and (0, 1) \to (-1, 0) show. So if the terminal point of x is (a, b), the terminal point of \displaystyle x + \frac{\pi}{2} is (-b, a), and its height is a:

The height of the sine graph at \displaystyle x + \frac{\pi}{2} equals the height of the cosine graph at x. Every point of the sine graph, slid \displaystyle \frac{\pi}{2} to the left, lands on the cosine graph. The cosine graph is the same wave, shifted, with the same rounded tops and steep crossings.

Facts about y = \cos x: domain all real numbers; range [-1, 1]; zeros at \displaystyle x = \frac{\pi}{2} + \pi n; highest points (x, 1) at x = 2\pi n; lowest points (x, -1) at x = \pi + 2\pi n; period 2\pi. It is symmetric about the y-axis, because the terminal points of x and -x are mirror images across the x-axis and have the same x-coordinate: \cos(-x) = \cos x, so cosine is an even function.

The five key points. Split one cycle into four quarter turns. Starting from 0, each quarter turn carries the terminal point from one axis to the next:

At these five angles every coordinate is 0, 1 or -1, so they are exactly where each graph crosses the axis or reaches its highest or lowest value. Between two neighbors the point stays in one quadrant, where both coordinates move steadily one way, so the graph only rises or only falls there, with the rounded shape found in Part 1.

Sine runs zero, top, zero, bottom, zero. Cosine runs top, zero, bottom, zero, top. To sketch a cycle by hand, plot the five points and join them with the rounded wave, never with straight segments. A cycle may start at any multiple of \displaystyle \frac{\pi}{2}: take that start and four more steps of \displaystyle \frac{\pi}{2}, and the five points are again the axis crossings and extremes.

Worked examples

Boundaries

• Radians. The input is in radians. In degrees the graph has the same up and down pattern, but the axis is marked 90^\circ, 180^\circ, \dots and the period is 360^\circ, so the curve is stretched sideways by a factor \displaystyle \frac{180}{\pi}. One axis cannot mix the two.
• Only these two graphs. Everything here is for y = \sin x and y = \cos x exactly. Steps of \displaystyle \frac{\pi}{2} between key points belong to period 2\pi; a number inside, as in \sin 2x, or in front, as in 3\cos x, changes the step or the heights, and later lessons handle each change.
• The letter x. On the graph, x is the angle. The x-coordinate of the terminal point is \cos x, a different number.
• Heights outside [-1, 1]. The graph never goes above 1 or below -1, so an equation such as \sin x = c or \cos x = c has no solution when |c| > 1.

Where this goes next

The two graphs drawn here are the parents of every sinusoid. Multiplying by a number stretches the wave up and down, which is Amplitude and reflection. A number multiplying x squeezes or spreads the cycle, the subject of Finding the period from the equation. Sliding the wave sideways is a phase shift, read correctly only after factoring the inside, and sliding it up or down moves the midline in Midline and vertical shift. Each of those moves the same five key points found here. Read in reverse, they let you write an equation from a wave, and they are the starting point for modeling things that repeat, from tides to the turning wheel. The same unrolling, done with the slope of the terminal side, gives the tangent graph and its asymptotes, and reading where one graph sits above a line is the heart of Trigonometric inequalities.

Exercises

Part 1: points on the sine graph

Give the exact point of the graph of y = \sin x at each x, and the quadrant of the terminal point.

1. \displaystyle x = -\frac{3\pi}{4}
2. \displaystyle x = \frac{13\pi}{6}
3. \displaystyle x = -\frac{5\pi}{3}

Part 1: reading the sine graph

4. Where on [0, 4\pi] is \sin x negative?
5. Where on [0, 4\pi] is y = \sin x decreasing?
6. Find every x in [-2\pi, 4\pi] with \sin x = -1.
7. Find every x in [0, 4\pi] with \displaystyle \sin x = -\frac{\sqrt{3}}{2}.
8. List the zeros of y = \sin x on [-3\pi, 3\pi]. How many are there?

Part 1: the reflector

Use Vihaan’s wheel from Example 4: the reflector is 1 foot from the hub, starts at (1, 0), and has height \sin x feet after a counterclockwise turn of x radians.

9. How high or low is the reflector after the wheel turns 4 radians, to the nearest hundredth of a foot, and in which quadrant is it?
10. For which x in [0, 4\pi] is the reflector at least \displaystyle \frac{\sqrt{2}}{2} foot above the hub?

Part 2: the cosine graph

11. Give the exact point of the graph of y = \cos x at \displaystyle x = \frac{7\pi}{6}.
12. Give the exact point of the graph of y = \cos x at \displaystyle x = -\frac{4\pi}{3}.
13. Where on [0, 2\pi] is y = \cos x increasing?
14. Find every x in [-2\pi, 2\pi] with \cos x = 0.
15. Find every x in [-2\pi, 2\pi] with \displaystyle \cos x = \frac{\sqrt{2}}{2}.

Part 2: key points of a cycle

16. Give the five key points of the cycle of y = \cos x that starts at x = \pi.
17. Give the five key points of the cycle of y = \sin x that starts at \displaystyle x = \frac{\pi}{2}, and say how this cycle begins.

Part 2: sine and cosine together

18. Where on [0, 2\pi] are \sin x and \cos x both negative?
19. Find every x in [0, 2\pi] with \sin x = -\cos x.
20. Meera turns Vihaan’s wheel through 15 radians from the start. After the start, how many times is the reflector level with the hub, how many times is it at the bottom, and how many times is it farthest back?

Reasoning

21. Using the motion of the terminal point, explain why the sine graph is steepest where it crosses the x-axis and flat at its highest and lowest points.
22. Show that \displaystyle \cos\left(x - \frac{\pi}{2}\right) = \sin x, and say what it means for the two graphs.
23. Show that the graph of y = \cos x is symmetric about the vertical line x = \pi.
24. Dhruv sketches y = \sin x on [0, 2\pi] by joining its five key points with straight segments. Meera says the sketch is wrong between the points. Use \displaystyle x = \frac{\pi}{6} to decide who is right.
25. Explain why the line y = c never meets the sine graph when |c| > 1, and why it meets it exactly twice in every cycle [2\pi n, 2\pi n + 2\pi) when -1 < c < 1.