Angles and the Unit Circle

You have been measuring angles your whole life. Every time you have estimated how far to turn a doorknob, judged whether a picture frame is level, or watched the hands of a clock sweep around its face, you have been working with angles. The unit circle simply takes this everyday intuition and puts it on a coordinate system, giving us a powerful way to connect geometry with algebra. By the end of this chapter, you will see that trigonometry is not some abstract set of rules to memorize; it is a natural extension of ideas you already understand.

Core Concepts

Angle Measurement: A Quick Review

An angle measures the amount of rotation between two rays (or line segments) that share a common endpoint, called the vertex. You are probably most familiar with measuring angles in degrees, where a full rotation around a circle is 360 degrees (written 360°).

Why 360? The ancient Babylonians chose this number, likely because it is close to the number of days in a year and has many divisors (making fractions easier to work with). Whatever its origin, this system stuck, and you have been using it ever since.

Here are the benchmark angles you already know:

• 90° is a right angle (a quarter turn, like the corner of a page)
• 180° is a straight angle (a half turn, like doing an about-face)
• 270° is three-quarters of a turn
• 360° is a full rotation (you end up where you started)

Standard Position

When we study angles mathematically, we place them in standard position:

• The vertex sits at the origin (0, 0)
• The initial side lies along the positive x-axis
• The terminal side is where the angle “ends” after rotating

Positive angles rotate counterclockwise (the way most people naturally draw a circle). Negative angles rotate clockwise.

This might seem like an arbitrary convention, but it gives us a consistent way to talk about angles so that everyone is working from the same reference point.

Radians: The Natural Way to Measure Angles

Degrees work perfectly well for everyday life, but mathematicians discovered a more natural unit for measuring angles: the radian.

Here is the key idea: imagine you have a circle with radius $r$. If you travel along the edge of the circle (the arc) for a distance equal to the radius itself, the angle you have swept out is exactly one radian.

$$1 \text{ radian} = \text{the angle where arc length equals radius}$$

Why is this “natural”? Because it connects angle measurement directly to the geometry of the circle itself. The radian is defined purely in terms of the circle, without any arbitrary choice like 360.

Since the circumference of a circle is $2\pi r$ (that is, $2\pi$ times the radius), a full rotation equals $2\pi$ radians. This gives us the fundamental conversion:

$$360° = 2\pi \text{ radians}$$

Or, simplifying:

$$180° = \pi \text{ radians}$$

Converting Between Degrees and Radians

To convert degrees to radians, multiply by $\frac{\pi}{180}$:

$$\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}$$

To convert radians to degrees, multiply by $\frac{180}{\pi}$:

$$\theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi}$$

Think of it this way: you are multiplying by a fraction that equals 1 (since $\pi$ radians = 180°), just expressed in the units you want to end up with.

Arc Length: How Far Around the Circle?

Once you understand radians, calculating arc length becomes beautifully simple. The arc length $s$ of a portion of a circle is:

$$s = r\theta$$

where $r$ is the radius and $\theta$ is the central angle in radians.

This formula works because of how radians are defined. If $\theta = 1$ radian, then $s = r$ (the arc length equals the radius). If $\theta = 2$ radians, then $s = 2r$ (the arc length is twice the radius). The relationship is perfectly linear.

Important: This formula only works when $\theta$ is in radians. If you are given an angle in degrees, convert it first.

Area of a Sector

A sector is a “pizza slice” of a circle (the region bounded by two radii and an arc). The area of a sector is:

$$A = \frac{1}{2}r^2\theta$$

where again $\theta$ must be in radians.

Where does this come from? A full circle has area $\pi r^2$ and corresponds to an angle of $2\pi$ radians. The fraction of the circle represented by angle $\theta$ is $\frac{\theta}{2\pi}$. So the sector area is:

$$A = \pi r^2 \times \frac{\theta}{2\pi} = \frac{1}{2}r^2\theta$$

The Unit Circle: Definition

The unit circle is simply a circle centered at the origin with radius 1. That is it. Nothing fancy. We call it “unit” because the radius is one unit.

Why radius 1? Because it makes everything simpler. In the arc length formula $s = r\theta$, when $r = 1$, the arc length simply equals the angle (in radians): $s = \theta$. This means we can measure how far we have traveled around the circle directly in radians.

Coordinates on the Unit Circle

Here is where the unit circle becomes powerful. Every point on the unit circle can be described by coordinates $(x, y)$. For a point at angle $\theta$ (measured from the positive x-axis):

$$x = \cos\theta$$ $$y = \sin\theta$$

This is actually the definition of cosine and sine. The cosine of an angle is the x-coordinate of the corresponding point on the unit circle. The sine is the y-coordinate.

Since every point on the unit circle is at distance 1 from the origin, and using the Pythagorean theorem:

$$x^2 + y^2 = 1$$ $$\cos^2\theta + \sin^2\theta = 1$$

This is the famous Pythagorean identity, and it follows directly from the definition of the unit circle.

Special Angles and Their Coordinates

Certain angles appear so frequently in mathematics that their coordinates are worth memorizing. These come from the familiar 30-60-90 and 45-45-90 triangles you may have encountered in geometry.

For a 45° angle (or $\frac{\pi}{4}$ radians), the point lies on the line $y = x$. Since $x^2 + y^2 = 1$ and $x = y$, we get $2x^2 = 1$, so $x = \frac{\sqrt{2}}{2}$. The coordinates are:

$$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$$

For 30° (or $\frac{\pi}{6}$ radians):

$$\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$$

For 60° (or $\frac{\pi}{3}$ radians):

$$\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$$

Notice that 30° and 60° are “partners”: the x-coordinate of one is the y-coordinate of the other.

Reference Angles

A reference angle is the acute angle (between 0° and 90°, or between 0 and $\frac{\pi}{2}$ radians) formed between the terminal side of an angle and the x-axis.

Reference angles let you use your knowledge of first-quadrant angles to find coordinates in any quadrant. No matter where your angle lands, you can relate it back to a familiar angle between 0° and 90°.

To find a reference angle:

• Quadrant I: The reference angle is just the angle itself
• Quadrant II: Reference angle = $180° - \theta$ (or $\pi - \theta$)
• Quadrant III: Reference angle = $\theta - 180°$ (or $\theta - \pi$)
• Quadrant IV: Reference angle = $360° - \theta$ (or $2\pi - \theta$)

Signs in Different Quadrants

The coordinates (and therefore sine and cosine) have different signs depending on which quadrant the terminal side lands in:

• Quadrant I (0° to 90°): Both $x$ and $y$ are positive, so $\cos\theta > 0$ and $\sin\theta > 0$
• Quadrant II (90° to 180°): $x$ is negative, $y$ is positive, so $\cos\theta < 0$ and $\sin\theta > 0$
• Quadrant III (180° to 270°): Both $x$ and $y$ are negative, so $\cos\theta < 0$ and $\sin\theta < 0$
• Quadrant IV (270° to 360°): $x$ is positive, $y$ is negative, so $\cos\theta > 0$ and $\sin\theta < 0$

Some people remember this with the mnemonic “All Students Take Calculus,” where the first letter of each word tells you what is positive in each quadrant (All, Sine, Tangent, Cosine), going counterclockwise from Quadrant I.

Coterminal Angles

Coterminal angles are angles that share the same terminal side. Since a full rotation is 360° (or $2\pi$ radians), adding or subtracting any multiple of 360° gives you a coterminal angle.

For example, 45°, 405°, and -315° are all coterminal. They look different as numbers, but they all point in the same direction.

To find coterminal angles:

$$\theta_{\text{coterminal}} = \theta + 360°n \quad \text{(or } \theta + 2\pi n \text{ in radians)}$$

where $n$ is any integer (positive, negative, or zero).

Notation and Terminology

Term	Symbol/Notation	Meaning
Degree	°	A unit of angle measure; 360° = one full rotation
Radian	rad (often omitted)	A unit of angle measure; $2\pi$ rad = one full rotation
Standard position	—	Vertex at origin, initial side on positive x-axis
Initial side	—	The starting ray of an angle (positive x-axis in standard position)
Terminal side	—	The ending ray after rotation
Arc length	$s$	Distance traveled along a circular arc; $s = r\theta$
Sector	—	A “pizza slice” region of a circle
Unit circle	—	Circle centered at origin with radius 1
Reference angle	$\theta’$ or $\theta_{\text{ref}}$	Acute angle to the x-axis
Coterminal	—	Angles sharing the same terminal side

Examples

Key Properties and Rules

Degree-Radian Conversions

Degrees	Radians
0°	0
30°	$\frac{\pi}{6}$
45°	$\frac{\pi}{4}$
60°	$\frac{\pi}{3}$
90°	$\frac{\pi}{2}$
120°	$\frac{2\pi}{3}$
135°	$\frac{3\pi}{4}$
150°	$\frac{5\pi}{6}$
180°	$\pi$
270°	$\frac{3\pi}{2}$
360°	$2\pi$

Unit Circle Coordinates (First Quadrant)

Angle (degrees)	Angle (radians)	Coordinates $(\cos\theta, \sin\theta)$
0°	0	$(1, 0)$
30°	$\frac{\pi}{6}$	$\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$
45°	$\frac{\pi}{4}$	$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
60°	$\frac{\pi}{3}$	$\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
90°	$\frac{\pi}{2}$	$(0, 1)$

Quadrant Signs Summary

Quadrant	Cosine (x)	Sine (y)
I	+	+
II	-	+
III	-	-
IV	+	-

Essential Formulas

• Arc length: $s = r\theta$ (with $\theta$ in radians)
• Sector area: $A = \frac{1}{2}r^2\theta$ (with $\theta$ in radians)
• Pythagorean identity: $\cos^2\theta + \sin^2\theta = 1$
• Coterminal angles: $\theta + 2\pi n$ (or $\theta + 360°n$) for any integer $n$

Real-World Applications

Navigation and Bearings

Pilots, sailors, and hikers use angles constantly. A compass bearing measures direction as an angle from north. When a pilot is told to fly at a heading of 270°, they know to fly due west. Understanding how angles work (including negative angles and angles beyond 360°) is essential for calculating routes and positions.

Rotating Machinery

Engineers who design gears, turbines, and motors work with angular velocity (how fast something rotates, measured in radians per second). The arc length formula helps calculate how far a point on a spinning disk or wheel travels. For example, knowing a car tire’s radius and how many radians it rotates tells you exactly how far the car has moved.

Computer Graphics and Animation

Video games and animated movies use the unit circle constantly. When a character turns or an object rotates on screen, the computer calculates new positions using coordinates from the unit circle. Every smooth rotation you see in a game involves trigonometry based on these principles.

Astronomy

Ancient astronomers used angles to track celestial bodies. Today, we still measure positions in the sky using angular coordinates. The arc length formula helps calculate distances when we know how far apart two objects appear (the angle) and their distance from us.

Architecture and Design

Architects use sectors when designing curved structures like domes, arches, and amphitheaters. Calculating the area of a sector helps determine how much material is needed for a curved roof section or a circular window.

Music and Sound

Sound waves are described using sine functions, which come directly from the unit circle. The angle in radians corresponds to the phase of the wave, and understanding how sine and cosine behave is essential for audio engineering and acoustics.

Self-Test Problems

Summary

• Degrees and radians are two ways to measure angles. A full rotation is 360° or $2\pi$ radians, so $180° = \pi$ radians.
• To convert degrees to radians, multiply by $\frac{\pi}{180}$. To convert radians to degrees, multiply by $\frac{180}{\pi}$.
• Arc length is given by $s = r\theta$ where $\theta$ is in radians. This formula works because radians are defined in terms of arc length.
• Sector area is given by $A = \frac{1}{2}r^2\theta$ where $\theta$ is in radians.
• The unit circle is centered at the origin with radius 1. Points on it have coordinates $(\cos\theta, \sin\theta)$.
• Special angles (30°, 45°, 60° and their multiples) have coordinates involving $\frac{1}{2}$, $\frac{\sqrt{2}}{2}$, and $\frac{\sqrt{3}}{2}$.
• A reference angle is the acute angle to the x-axis. Use it to find coordinates in any quadrant by adjusting signs appropriately.
• The signs of sine and cosine depend on the quadrant: remember “All Students Take Calculus” or simply think about which coordinates are positive in each quadrant.
• Coterminal angles share the same terminal side and differ by multiples of 360° (or $2\pi$ radians).
• These concepts form the foundation of trigonometry and appear throughout physics, engineering, computer graphics, and countless other fields.