Trigonometric Functions and Unit Circle

TL;DR

Trigonometric functions (sine, cosine, tangent) describe relationships between angles and side lengths in right triangles. The unit circle extends these ideas beyond just right triangles, allowing you to find trig values for any angle. Understanding the unit circle is key to graphing trig functions and solving many problems.

1. The Mental Model

Imagine a Ferris wheel with a radius of exactly 1. As you ride it, your height and how far you are from the center (horizontally) change with the angle you've rotated. This Ferris wheel is your unit circle, and your position directly relates to sine and cosine.

2. The Core Material

Trigonometry started with right triangles. You learned about SOH CAH TOA:
- Sine ($\sin$) = Opposite / Hypotenuse
- Cosine ($\cos$) = Adjacent / Hypotenuse
- Tangent ($\tan$) = Opposite / Adjacent

This is great for angles between 0 and 90 degrees. But what about angles like 120 or 300 degrees? That's where the unit circle comes in.

The Unit Circle

Photo by Jenny Uhling on Pexels

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

Imagine an angle, $\theta$, starting from the positive x-axis and rotating counter-clockwise. The point where the angle's "terminal side" (the ray forming the angle) intersects the unit circle is super important. Let's call this point (x, y).

• The x-coordinate of this point is the cosine of the angle ($\cos \theta = x$).
• The y-coordinate of this point is the sine of the angle ($\sin \theta = y$).

Why? Because if you drop a perpendicular from (x,y) to the x-axis, you form a right triangle with hypotenuse 1 (the radius). The adjacent side is x, and the opposite side is y. So, $\cos \theta = x/1 = x$ and $\sin \theta = y/1 = y$.

The tangent is then $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{y}{x}$.

Here's how the main components relate:

graph TD
    A["Angle (theta) from positive x-axis"] --> B["Point (x, y) on Unit Circle"]
    B --> C["x-coordinate is cos(theta)"]
    B --> D["y-coordinate is sin(theta)"]
    C & D --> E["tan(theta) = y/x"]
    B --> F["Radius is 1 (Hypotenuse)"]

Important Angles and Quadrants

Photo by DS stories on Pexels

The unit circle helps you find trig values for special angles (like 0, 30, 45, 60, 90 degrees and their multiples) without a calculator. You'll often see these in radians as well:
- 0 degrees = 0 radians
- 30 degrees = $\pi/6$ radians
- 45 degrees = $\pi/4$ radians
- 60 degrees = $\pi/3$ radians
- 90 degrees = $\pi/2$ radians

The circle is divided into four quadrants. The signs of sine, cosine, and tangent depend on the quadrant:
- Quadrant I (0 to 90 degrees or $0$ to $\pi/2$): x > 0, y > 0. All positive.
- Quadrant II (90 to 180 degrees or $\pi/2$ to $\pi$): x < 0, y > 0. Sine positive, cosine negative, tangent negative.
- Quadrant III (180 to 270 degrees or $\pi$ to $3\pi/2$): x < 0, y < 0. Tangent positive, sine negative, cosine negative.
- Quadrant IV (270 to 360 degrees or $3\pi/2$ to $2\pi$): x > 0, y < 0. Cosine positive, sine negative, tangent negative.

A common mnemonic is "All Students Take Calculus" to remember which functions are positive in which quadrant (starting from Q1 and going counter-clockwise).

3. Worked Example

Let's find the sine, cosine, and tangent for an angle of $210^\circ$.

1. Locate the angle: $210^\circ$ is in the third quadrant (between $180^\circ$ and $270^\circ$).
2. Find the reference angle: This is the acute angle formed between the terminal side of $210^\circ$ and the x-axis. For $210^\circ$, it's $210^\circ - 180^\circ = 30^\circ$.
3. Determine the coordinates for the reference angle: You should know (or be able to derive) that for $30^\circ$ on the unit circle, the coordinates are $(\cos 30^\circ, \sin 30^\circ) = (\sqrt{3}/2, 1/2)$.
4. Apply quadrant signs: Since $210^\circ$ is in Quadrant III, both the x and y coordinates (cosine and sine) will be negative.
   • So, for $210^\circ$, the point on the unit circle is $(-\sqrt{3}/2, -1/2)$.
5. State the trig values:
   • $\cos 210^\circ = -\sqrt{3}/2$
   • $\sin 210^\circ = -1/2$
   • $\tan 210^\circ = \frac{\sin 210^\circ}{\cos 210^\circ} = \frac{-1/2}{-\sqrt{3}/2} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

4. Key Takeaways

• The unit circle is a circle with a radius of 1, centered at the origin, used to define trigonometric functions for any angle.
• For an angle $\theta$ on the unit circle, the x-coordinate of the intersection point is $\cos \theta$, and the y-coordinate is $\sin \theta$.
• $\tan \theta = \sin \theta / \cos \theta$.
• Understanding the unit circle helps you quickly find exact trig values for common angles (multiples of $30^\circ$, $45^\circ$, $60^\circ$).
• The sign of sine, cosine, and tangent depends on the quadrant the angle terminates in.
• Reference angles are crucial for finding trig values of angles outside the first quadrant.

Common Mistakes to Avoid:
- Mixing up sine and cosine: remember $(\cos \theta, \sin \theta)$ is $(x, y)$.
- Forgetting to consider the sign based on the quadrant.
- Confusing degrees and radians; always pay attention to the units.
- Not knowing the common special angle values for $30^\circ, 45^\circ, 60^\circ$.

5. Now Try It

Without using a calculator, find the exact values for $\sin(3\pi/4)$, $\cos(3\pi/4)$, and $\tan(3\pi/4)$.

What success looks like: You should have three distinct, exact numerical values (possibly involving square roots or fractions) for sine, cosine, and tangent, and they should have the correct signs based on the quadrant.