Why the Sine, Cosine and Tangent Graphs Look the Way They Do — Cosine Is Sine, Shifted Sideways

Introduction

Open a textbook and three trigonometric graphs appear side by side: the smoothly rolling sine curve, the cosine curve that looks like that same wave nudged a little to the side, and — the one with a personality all its own — the tangent curve that suddenly breaks off and then leaps back up toward the sky. Many people memorize these three pictures whole: sine starts at the origin and rises, cosine starts at the top and falls, tangent breaks somewhere along the way.

But these shapes are not pictures to be memorized. All three flow naturally out of a single circle. Watch just one point going around a circle and you can see at a glance why sine and cosine look identical yet sit offset from each other, and why only tangent breaks apart. In this post we follow how that one point draws all three graphs.

One piece of terminology first. When we measure an angle, the ray with one end fixed at the origin and rotated by the given angle is called the terminal side. Like the second hand of a clock, a needle starting from an axis and turned by an angle \(\theta\) (theta, the Greek letter for an angle) is the terminal side, and the point where its tip meets the circle we will call \(P\).

1. Where the graph comes from — the two shadows of a point on the circle

On a circle of radius \(1\) (the unit circle), the terminal side turns by an angle \(\theta\) and meets the circle at a point \(P\). That point's horizontal coordinate is the cosine, its vertical coordinate the sine.

Here the true nature of the graph reveals itself. In the trig graphs we look at, the horizontal axis is the angle and the vertical axis is the function value at that angle. So if you turn the terminal side little by little and, at each moment, plot the vertical coordinate of \(P\) one step to the right, those points join into the sine curve. Plot the horizontal coordinate the same way and you get the cosine curve.

In plain terms: as the point \(P\) makes one loop around the circle, the graph is its shadow cast on a wall, laid out in order of time. Lay out the shadow on a vertical wall (its height) and you get sine; lay out the shadow on a horizontal wall (its left–right position) and you get cosine. The point merely spins around the circle, yet unrolling how its shadow rises and falls produces the wave shape.

This also makes it obvious why both curves oscillate between \(-1\) and \(1\). The point \(P\) cannot leave a circle of radius \(1\), so its shadow (its coordinate) cannot leave the range between \(-1\) and \(1\). Grow the angle from \(0°\) and the vertical shadow rises and falls \(0 \to 1 \to 0 \to -1 \to 0\); unrolling that rise and fall is exactly the sine wave we know.

2. Cosine is sine pushed sideways

Now overlay the two curves and an interesting fact appears: sine and cosine are completely identical in shape and differ only in position. Sine starts from \(0\) at \(0°\) and reaches its peak (\(1\)) at \(90°\), while cosine is already at its peak (\(1\)) at \(0°\) and falls to \(0\) at \(90°\). Cosine is running exactly \(90°\) ahead of sine.

Why \(90°\), of all values? Go back to the circle and the answer appears at once. The horizontal coordinate of \(P\) (cosine) and its vertical coordinate (sine) are shadows in two directions that are \(90°\) apart. Just when the point has gone furthest horizontally (its horizontal shadow is at maximum), it still has a long way to climb vertically — it must turn exactly \(90°\) more before the vertical shadow reaches its maximum. Because the two shadows repeat the same rise and fall with a \(90°\) lag between them, the two curves are the same wave pushed over by \(90°\).

Writing this in a single equation:

In words: "the cosine of an angle equals the sine of that angle plus \(90°\)." On the graph, it says that pushing the cosine curve to the right by \(90°\) makes it land exactly on the sine curve. In the interactive below, shift cosine over by \(90°\) onto sine and you can watch the two curves merge into one without the slightest gap.

So sine and cosine are really two faces of one curve. The only difference is whether you watch the vertical shadow or the horizontal shadow of the point going around the circle.

3. Why the tangent graph breaks apart

If sine and cosine are gentle waves, tangent has an entirely different temperament. The tangent curve climbs from below, then at some moment shoots up into the sky and vanishes, only to reappear from beneath the ground a while later. It is chopped into pieces in the middle. Why is it shaped this way?

The key is that tangent is not a single coordinate but a ratio of the two coordinates.

Sine and cosine read a shadow off directly, but tangent divides the vertical shadow by the horizontal one. The moment division enters, things change. If the divisor (the horizontal shadow, i.e. cosine) approaches \(0\), the quotient grows out of control, and if it hits exactly \(0\), the value ceases to exist at all.

Where does the horizontal shadow become \(0\)? When the point \(P\) is at the very top of the circle (\(90°\)) or the very bottom (\(270°\)). There the point clings to the vertical axis and its horizontal coordinate is \(0\). So as the angle approaches \(90°\) or \(270°\), tangent shoots to infinity, and at those exact spots you divide by \(0\) and there is simply no value. On the graph a vertical boundary line appears there that the curve can never touch, and this is called an asymptote — a line the curve approaches endlessly but never meets.

That is why the tangent curve, broken at each asymptote, repeats the same shape every \(180°\). (How tangent flips sign as it shoots up near \(90°\) is covered in more detail in Why Does Only tan Blow Up to Infinity.) In exact opposition to sine and cosine, which cannot leave the circle and so stay trapped in a narrow band, tangent — carrying a division inside — bursts through that band and opens up and down. What split the character of the three graphs was, in the end, whether you look at a coordinate or at a ratio of coordinates.

4. See it for yourself

In the interactive below, sweep the \(\theta\) slider from \(0°\) to \(360°\). As the point \(P\) turns around the left circle, its horizontal coordinate (cosine, green) and vertical coordinate (sine, blue) are plotted as waves on the graph at the right. You can see with your own eyes that the two waves are identical in shape and merely offset in position.

Press the 'shift cos right by 90° to overlap sine' button and the shifted cosine (pink dashed) lands exactly on the sine curve — showing the two are one curve. Press the 'show the tan graph' button and the tangent curve (amber) is drawn as well; you can watch it break at the asymptotes at \(90°\) and \(270°\) and flee up and down.

Key takeaways

• The trig graphs are not pictures to memorize but the shadows of a point on the unit circle, unrolled in order of time. Unroll the vertical shadow and you get the sine curve; unroll the horizontal shadow and you get the cosine curve.
• The two shadows are one body offset by \(90°\), so cosine is sine pushed sideways: \(\cos\theta = \sin(\theta + 90°)\). Sine and cosine are two faces of one curve.
• Tangent is not a coordinate but a ratio of two coordinates (vertical ÷ horizontal), so at \(90°\) and \(270°\), where the horizontal shadow (cosine) becomes \(0\), it shoots to infinity and loses its value (an asymptote). That is why the curve, chopped into pieces, repeats every \(180°\).

A single point going around a circle, and the shadow it casts on a wall — that was everything about the three curves. The gentle wave, its sideways-pushed twin, and the curve broken into pieces are, in the end, just what you see when you look in different directions at the same circle.

Related reading — A Right Triangle Only Reaches 90° — So How Do We Measure sin 120°? (redefining the ratios as coordinates on a circle — the starting point of these graphs) · Why Does Only tan Blow Up to Infinity (a deeper look at why tangent explodes at its asymptotes) · Why Do −60° and 300° Have the Same Sine? (trig functions seen as direction — the periodicity that makes the graphs repeat)