5.2 Unit Circle: Sine and Cosine Functions

Unit Circle and Trigonometric Functions

The unit circle gives you a visual framework for understanding sine and cosine. Every point on the circle corresponds to an angle, and the coordinates of that point are the cosine and sine values. Once you learn how this works for a few key angles, reference angles let you extend that knowledge to any angle on the circle.

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Sine and cosine for common angles

The unit circle is a circle with radius 1, centered at the origin $(0, 0)$. Angles are measured counterclockwise from the positive x-axis. For any angle $\theta$, the point where the terminal side intersects the unit circle has coordinates $(\cos\theta, \sin\theta)$.

That's the core idea: the x-coordinate is cosine, and the y-coordinate is sine.

Three angles show up constantly in pre-calc and beyond. You should know their radian equivalents and their unit circle coordinates cold:

• 30° = $\frac{\pi}{6}$ → point on circle: $\left(\frac{\sqrt{3}}{2},\; \frac{1}{2}\right)$
  • $\cos\!\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$, $\sin\!\left(\frac{\pi}{6}\right) = \frac{1}{2}$
• 45° = $\frac{\pi}{4}$ → point on circle: $\left(\frac{\sqrt{2}}{2},\; \frac{\sqrt{2}}{2}\right)$
  • $\cos\!\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$, $\sin\!\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$
• 60° = $\frac{\pi}{3}$ → point on circle: $\left(\frac{1}{2},\; \frac{\sqrt{3}}{2}\right)$
  • $\cos\!\left(\frac{\pi}{3}\right) = \frac{1}{2}$, $\sin\!\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$

A useful pattern: the sine values for 30°, 45°, and 60° follow $\frac{\sqrt{1}}{2}$, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{3}}{2}$. The cosine values go in reverse order. Noticing this makes memorization much easier.

Don't forget the quadrantal angles either: at 0° the point is $(1, 0)$, at 90° it's $(0, 1)$, at 180° it's $(-1, 0)$, and at 270° it's $(0, -1)$.

Domain and range of trigonometric functions

Because you can rotate an angle as far as you want in either direction, the domain of both sine and cosine is all real numbers: $(-\infty, \infty)$.

The range is more restricted. Since every point on the unit circle satisfies $x^2 + y^2 = 1$, neither coordinate can be greater than 1 or less than $-1$. So for both functions:

$-1 \leq \sin(\theta) \leq 1 \qquad \text{and} \qquad -1 \leq \cos(\theta) \leq 1$

• Sine reaches its maximum of 1 at $90°$ ($\frac{\pi}{2}$) and its minimum of $-1$ at $270°$ ($\frac{3\pi}{2}$).
• Cosine reaches its maximum of 1 at $0°$ (and $360°$) and its minimum of $-1$ at $180°$ ($\pi$).

Both sine and cosine are periodic functions, meaning they repeat their values in a regular cycle. The period is $2\pi$ radians (360°), so $\sin(\theta + 2\pi) = \sin(\theta)$ for any $\theta$, and the same holds for cosine.

Applying Reference Angles

Reference angles in the unit circle

A reference angle is the acute angle (between 0° and 90°) formed between the terminal side of your angle and the x-axis. It's always measured to the nearest portion of the x-axis, not the y-axis.

To find a reference angle:

• Quadrant I (0° to 90°): reference angle = the angle itself
• Quadrant II (90° to 180°): reference angle = $180° - \theta$
• Quadrant III (180° to 270°): reference angle = $\theta - 180°$
• Quadrant IV (270° to 360°): reference angle = $360° - \theta$

The reference angle tells you the magnitude of sine and cosine. The quadrant tells you the sign. A common mnemonic is "All Students Take Calculus" (ASTC), going counterclockwise from Quadrant I:

1. Quadrant I — All trig functions positive
2. Quadrant II — Sine positive, cosine negative
3. Quadrant III — Tangent positive (sine and cosine both negative)
4. Quadrant IV — Cosine positive, sine negative

Using reference angles step by step

Here's the process for finding sine or cosine of any angle:

1. Identify the quadrant the angle falls in.
2. Calculate the reference angle using the formulas above.
3. Look up sine and cosine of the reference angle (it'll be one of the common angles you've memorized).
4. Apply the correct signs based on the quadrant.

Example: Find $\sin(210°)$ and $\cos(210°)$.

1. 210° is between 180° and 270°, so it's in Quadrant III.

2. Reference angle: $210° - 180° = 30°$.

3. From the unit circle: $\sin(30°) = \frac{1}{2}$, $\cos(30°) = \frac{\sqrt{3}}{2}$.

4. In Quadrant III, both sine and cosine are negative:

$\sin(210°) = -\frac{1}{2}, \qquad \cos(210°) = -\frac{\sqrt{3}}{2}$

Example: Find $\sin(315°)$ and $\cos(315°)$.

1. 315° is in Quadrant IV.

2. Reference angle: $360° - 315° = 45°$.

3. $\sin(45°) = \frac{\sqrt{2}}{2}$, $\cos(45°) = \frac{\sqrt{2}}{2}$.

4. In Quadrant IV, sine is negative and cosine is positive:

$\sin(315°) = -\frac{\sqrt{2}}{2}, \qquad \cos(315°) = \frac{\sqrt{2}}{2}$

Key connections to other topics

A few concepts build directly on the unit circle and will come up later in the course:

• Pythagorean identity: Since any point on the unit circle satisfies $x^2 + y^2 = 1$, you get $\cos^2(\theta) + \sin^2(\theta) = 1$. This is the most fundamental trig identity.
• Periodic behavior: The repeating nature of sine and cosine connects to graphing trig functions, where you'll study amplitude, period, and phase shifts.
• Parametric equations: The unit circle can be described parametrically as $x = \cos(t)$, $y = \sin(t)$, which you'll use to model circular and periodic motion.