Introduction to the Unit Circle in Trigonometry

Mathematically, for any point $(x, y)$ on the unit circle:

$$\begin{aligned} x &= \cos(\theta) \\ y &= \sin(\theta) \\ \tan(\theta) &= \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)} \end{aligned}$$

This mapping is revolutionary because it allows the functions to take on negative values. As the point moves into the second, third, or fourth quadrants, the $x$ and $y$ coordinates change signs, allowing $\cos(\theta)$ and $\sin(\theta)$ to represent values below zero or to the left of the origin.

Exam Tip: Always remember that $\cos$ is associated with $x$ (horizontal) and $\sin$ is associated with $y$ (vertical). A common mistake is to swap them when calculating coordinates in different quadrants.

The Pythagorean Identity

Because every point $(x, y)$ on the unit circle forms a right triangle with the $x$-axis, we can apply the Pythagorean Theorem ($a^2 + b^2 = c^2$). In this context, the legs of the triangle are the $x$ and $y$ coordinates, and the hypotenuse is the radius $r = 1$.

Substituting our trigonometric definitions into the Pythagorean Theorem gives us the most fundamental identity in trigonometry:

$$\begin{aligned} x^2 + y^2 &= 1^2 \\ (\cos \theta)^2 + (\sin \theta)^2 &= 1 \\ \cos^2 \theta + \sin^2 \theta &= 1 \end{aligned}$$

This identity is a powerful tool for solving equations. If you know the value of $\sin \theta$ and need to find $\cos \theta$, you don't need to know the angle itself; you can simply use this relationship to solve for the missing coordinate.

Worked Example: Finding Values in Quadrant II

Let's apply these concepts to a practical problem. Suppose we are given that $\sin \theta = \frac{\sqrt{3}}{2}$ and the angle $\theta$ is in the second quadrant ($90^\circ < \theta < 180^\circ$). We want to find $\cos \theta$ and $\tan \theta$.

Step 1: Use the Pythagorean Identity to find $\cos \theta$

$$\begin{aligned} \cos^2 \theta + \sin^2 \theta &= 1 \\ \cos^2 \theta + \left(\frac{\sqrt{3}}{2}\right)^2 &= 1 \\ \cos^2 \theta + \frac{3}{4} &= 1 \\ \cos^2 \theta &= 1 - \frac{3}{4} \\ \cos^2 \theta &= \frac{1}{4} \end{aligned}$$

Step 2: Solve for $\cos \theta$ and consider the quadrant Taking the square root of both sides gives $\cos \theta = \pm \frac{1}{2}$. Since we are in the second quadrant, the $x$-coordinate must be negative. Therefore: $\cos \theta = -\frac{1}{2}$

Step 3: Find $\tan \theta$ Using the ratio of sine to cosine:

$$\begin{aligned} \tan \theta &= \frac{\sin \theta}{\cos \theta} \\ \tan \theta &= \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} \\ \tan \theta &= -\sqrt{3} \end{aligned}$$

Summary Table of Quadrant Signs

To navigate the unit circle efficiently, it is helpful to memorize which trigonometric functions are positive in each quadrant:

By mastering the relationship between the $(x, y)$ coordinates and the trigonometric ratios, you move from simply memorizing triangles to understanding the continuous, circular nature of mathematics.