The unit circle is a super important idea in trigonometry. It's especially useful when we start to look at how angles connect to trigonometric functions.

So, what is the unit circle?

It's simply a circle with a radius of 1. This circle is centered right in the middle of a grid called a coordinate plane. The unit circle makes it easier to understand the values of sine, cosine, and tangent.

Key Points on the Unit Circle

The unit circle has some key points that are linked to specific angles. We usually measure these angles in radians. Here are some important ones:

• $0^\circ$ (or $0$ radians): The point is $(1, 0)$. Here, $\cos(0) = 1$ and $\sin(0) = 0$.

• $90^\circ$ (or $\frac{\pi}{2}$ radians): The point is $(0, 1)$. So, $\cos\left(\frac{\pi}{2}\right) = 0$ and $\sin\left(\frac{\pi}{2}\right) = 1$.

• $180^\circ$ (or $\pi$ radians): The point is $(-1, 0)$. This means $\cos(\pi) = -1$ and $\sin(\pi) = 0$.

• $270^\circ$ (or $\frac{3\pi}{2}$ radians): The point is $(0, -1)$. Thus, $\cos\left(\frac{3\pi}{2}\right) = 0$ and $\sin\left(\frac{3\pi}{2}\right) = -1$.

• $360^\circ$ (or $2\pi$ radians): The point goes back to $(1, 0)$.

How Trigonometric Functions Work with the Unit Circle

From these key points, we can see how trigonometric functions fit with the unit circle:

• Cosine is linked to the $x$-coordinate of a point on the circle.

• Sine is linked to the $y$-coordinate of that same point.

• Tangent is found by dividing sine by cosine:

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

This connection shows us how angles on the unit circle help us find the values for sine, cosine, and tangent easily. By moving around the unit circle, we can discover the sine, cosine, and tangent values for any angle!