Sine and cosine are coordinates

Stand on the unit circle from last lesson. Pick an angle $\theta$. Rotate the point $(1, 0)$ counterclockwise by $\theta$. It lands somewhere on the circle, at some coordinates $(x, y)$.

Those coordinates have names:

That is the definition. Cosine is the $x$-coordinate of the point. Sine is the $y$-coordinate. Nothing is happening here except naming the two numbers that locate a point on a circle.

Drag the point. The horizontal segment is $\cos\theta$, the vertical drop is $\sin\theta$. They are the coordinates, live.

Because $\cos\theta$ and $\sin\theta$ are just coordinates, their signs are not a rule to memorize. They are wherever the point is.

In the first quadrant both are positive. In the second quadrant the point is up and to the left, so $x < 0$ and $y > 0$: cosine negative, sine positive. Third quadrant, both negative. Fourth, $x > 0$ and $y < 0$.

Do not memorize a sign chart. Drag the point into each quadrant and read the signs off the axes. The picture is the chart.

Most people meet sine and cosine first as triangle ratios, months before they ever see a circle. Those two stories get filed as separate topics. They are not. They are the same picture.

Take the point $(\cos\theta, \sin\theta)$ on the unit circle and drop a straight vertical line from it to the $x$-axis. You have just drawn a right triangle:

• the hypotenuse is the radius, running from the origin to the point, length $1$,
• the opposite leg is the vertical drop, length $\sin\theta$,
• the adjacent leg lies along the $x$-axis, length $\cos\theta$.

The triangle was inside the circle the whole time. Its legs are literally the coordinates.

Now the part that makes the two stories one story.

In the widget, drag the vertex outward so the hypotenuse grows past length $1$. The triangle gets bigger. The leg lengths grow to $r\sin\theta$ and $r\cos\theta$, where $r$ is the new hypotenuse length.

But watch the two ratios in the readout:

The $r$ cancels. Stretch the triangle as much as you like: the leg lengths change, but those two ratios do not budge. They depend only on the angle.

That is the school mnemonic, and now it is a consequence instead of a starting point:

It works for any right triangle, not just ones with hypotenuse $1$, precisely because the ratio cancels the scale. The unit circle is just the case where the hypotenuse is $1$ and the ratios are the legs themselves.

One definition. The circle is the clean version; the triangle is what you get when the hypotenuse is some other length $r$.

There is one more combination worth a name. Define

Two ways to see it. As a triangle ratio, dividing $\text{opposite}/\text{hypotenuse}$ by $\text{adjacent}/\text{hypotenuse}$ cancels the hypotenuse and leaves $\text{opposite}/\text{adjacent}$.

As a circle fact, $\tan\theta$ is the slope of the radius drawn to $(\cos\theta, \sin\theta)$: rise over run is $\sin\theta / \cos\theta$. Tangent is steepness. It blows up toward infinity as $\theta$ approaches $90^\circ$, exactly when the radius goes vertical and the run hits zero.

Here is the most-quoted identity in trigonometry, and it costs you nothing.

The point $(\cos\theta, \sin\theta)$ is on the unit circle. The unit circle is the set where $x^2 + y^2 = 1$. Substitute $x = \cos\theta$ and $y = \sin\theta$:

That is it. It is not a trick pulled out of nowhere; it is the Pythagorean theorem with the legs renamed. Watch the sin²θ + cos²θ readout in the circle widget as you drag: it sits at $1.000$ forever, because the point never leaves the circle.

The identity is a labor-saver. If you know one of $\sin\theta$ or $\cos\theta$, you know the other up to a sign, and the quadrant fixes the sign.

From here on, when this course says $\sin$ or $\cos$, it means the unit-circle coordinate. The triangle picture is the special case at hypotenuse $1$, and you can always summon it by dropping a vertical line.

So far $\theta$ has been an angle. Next lesson we cut that cord. We feed $\sin$ and $\cos$ any real number at all and watch what their outputs draw. The answer is a wave, and that wave is the shape a transformer will later use to tell positions apart.