Unit Circle and Radian Measure

In early geometry, trigonometry is often limited to right triangles, which means we only deal with acute angles (between $0^\circ$ and $90^\circ$). By using the unit circle and radian measure, we can extend trigonometric functions to evaluate any angle—including negative angles and angles greater than a full rotation.

What is a Radian?

A radian is a unit of angle measure based on the radius of a circle. One radian is the angle created when the arc length along the circle is exactly equal to the length of the radius.

Since the circumference of a full circle is $2\pi r$, a full $360^\circ$ rotation is equal to $2\pi$ radians.

This gives us the fundamental conversion: $180^\circ = \pi \text{ radians}$

• Degrees to Radians: Multiply by $\frac{\pi}{180^\circ}$
• Radians to Degrees: Multiply by $\frac{180^\circ}{\pi}$

The Unit Circle

The unit circle is simply a circle with a radius of $1$ centered at the origin $(0,0)$ of a coordinate plane.

For any angle $\theta$ drawn in standard position (starting from the positive x-axis), the point where the terminal side of the angle intersects the unit circle has the coordinates: $(x, y) = (\cos \theta, \sin \theta)$

Because the radius is $1$, the Pythagorean theorem ($x^2 + y^2 = r^2$) gives us the fundamental trigonometric identity: $\cos^2 \theta + \sin^2 \theta = 1$

Navigating the Four Quadrants

Angles can rotate counterclockwise (positive angles) or clockwise (negative angles). You can also rotate multiple times, meaning angles can be greater than $360^\circ$ or $2\pi$. The coordinate signs in the four quadrants determine the signs of your trigonometric functions (often remembered by the acronym CAST):

• Quadrant I ($0$ to $\frac{\pi}{2}$): Both $x$ and $y$ are positive. All trig functions are positive.
• Quadrant II ($\frac{\pi}{2}$ to $\pi$): $x$ is negative, $y$ is positive. Only Sine is positive.
• Quadrant III ($\pi$ to $\frac{3\pi}{2}$): Both $x$ and $y$ are negative. Only Tangent ($\frac{\sin}{\cos}$) is positive.
• Quadrant IV ($\frac{3\pi}{2}$ to $2\pi$): $x$ is positive, $y$ is negative. Only Cosine is positive.

Reference Angles and Special Values

To find the trig value of any large or negative angle, find its reference angle—the acute angle made with the x-axis. The trig values for $\theta$ will match the values of its reference angle, with the sign adjusted based on the quadrant.

Memorizing the first-quadrant values for special angles is crucial:

• $\frac{\pi}{6}$ ($30^\circ$): $\cos = \frac{\sqrt{3}}{2}$, $\sin = \frac{1}{2}$
• $\frac{\pi}{4}$ ($45^\circ$): $\cos = \frac{\sqrt{2}}{2}$, $\sin = \frac{\sqrt{2}}{2}$
• $\frac{\pi}{3}$ ($60^\circ$): $\cos = \frac{1}{2}$, $\sin = \frac{\sqrt{3}}{2}$

Example Problems

Example 1: Find the exact value of $\sin\left(\frac{5\pi}{6}\right)$

1. Locate the angle: $\frac{5\pi}{6}$ is slightly less than $\pi$, placing it in Quadrant II.
2. Find the reference angle: The distance to the x-axis ($\pi$) is $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$.
3. Determine the sign: In Quadrant II, sine (the y-coordinate) is positive.
4. Evaluate: $\sin\left(\frac{5\pi}{6}\right) = +\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$.

Example 2: Find all angles $\theta$ in $[0, 2\pi)$ where $\cos \theta = -\frac{\sqrt{3}}{2}$

1. Find the reference angle: We know that $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$. So, our reference angle is $\frac{\pi}{6}$.
2. Determine the quadrants: Cosine (the x-coordinate) is negative in Quadrants II and III.
3. Calculate the angles:
   • Quadrant II: $\pi - \frac{\pi}{6} = \frac{5\pi}{6}$
   • Quadrant III: $\pi + \frac{\pi}{6} = \frac{7\pi}{6}$

The solutions are $\theta = \frac{5\pi}{6}$ and $\theta = \frac{7\pi}{6}$.