📚 Table of Contents
- 1. What is an Angle?
- 2. Degrees vs Radians
- 3. The Unit Circle
- 4. Trig Values on the Unit Circle
- 5. Reference Angles
- 6. The Six Trig Functions
- 7. Points on the Unit Circle
- 8. Arc Length & Sector Area
- 9. Extra Practice
1. What is an Angle?
An angle is formed by two rays sharing a common endpoint called the vertex.
- Standard Position: vertex at the origin, initial side along the positive x-axis
- Positive angles: rotate counterclockwise ↺
- Negative angles: rotate clockwise ↻
- Coterminal angles: different angles that land on the same terminal side
To find coterminal angles, add or subtract 360° (or \(2\pi\)):
Coterminal Angle \(= \theta \pm 360°\) or \(\theta \pm 2\pi\)
✏️ Example: Find a positive coterminal angle for \(-60°\)
\(-60° + 360° =\) 300°
2. Degrees vs Radians
Angles can be measured in degrees or radians. Radians are used most in higher math.
| Degrees | Radians |
|---|---|
| 0° | \(0\) |
| 30° | \(\pi/6\) |
| 45° | \(\pi/4\) |
| 60° | \(\pi/3\) |
| 90° | \(\pi/2\) |
| 180° | \(\pi\) |
| 270° | \(3\pi/2\) |
| 360° | \(2\pi\) |
Conversion Formulas:
- Degrees → Radians: multiply by \(\dfrac{\pi}{180}\)
- Radians → Degrees: multiply by \(\dfrac{180}{\pi}\)
✏️ Example 1: Convert 135° to radians
$$135 \times \frac{\pi}{180} = \frac{3\pi}{4}$$✏️ Example 2: Convert \(\dfrac{5\pi}{6}\) to degrees
$$\frac{5\pi}{6} \times \frac{180}{\pi} = 150°$$3. The Unit Circle
The unit circle is a circle with radius = 1 centered at the origin.
Every point on the unit circle is written as \((\cos\theta,\ \sin\theta)\) where \(\theta\) is the angle.
Key idea: The unit circle lets you find exact trig values for any standard angle.
✏️ Example: What point on the unit circle corresponds to \(\theta = 60°\)?
\(\theta = 60°\) → point \(= \left(\cos 60°,\ \sin 60°\right) = \left(\dfrac{1}{2},\ \dfrac{\sqrt{3}}{2}\right)\)
4. Trig Values on the Unit Circle
| Angle (°) | Angle (rad) | \(\sin\theta\) | \(\cos\theta\) | \(\tan\theta\) |
|---|---|---|---|---|
| 0° | \(0\) | \(0\) | \(1\) | \(0\) |
| 30° | \(\pi/6\) | \(1/2\) | \(\sqrt{3}/2\) | \(\sqrt{3}/3\) |
| 45° | \(\pi/4\) | \(\sqrt{2}/2\) | \(\sqrt{2}/2\) | \(1\) |
| 60° | \(\pi/3\) | \(\sqrt{3}/2\) | \(1/2\) | \(\sqrt{3}\) |
| 90° | \(\pi/2\) | \(1\) | \(0\) | undefined |
| 180° | \(\pi\) | \(0\) | \(-1\) | \(0\) |
| 270° | \(3\pi/2\) | \(-1\) | \(0\) | undefined |
| 360° | \(2\pi\) | \(0\) | \(1\) | \(0\) |
✏️ Example: Find the exact value of \(\sin(225°)\)
- 225° is in Quadrant III
- Reference angle \(= 225° – 180° = 45°\)
- \(\sin\) is negative in Q III
- \(\sin(225°) = -\dfrac{\sqrt{2}}{2}\)
5. Reference Angles
A reference angle is the acute angle (between 0° and 90°) formed between the terminal side and the x-axis.
| Quadrant | Reference Angle Formula | \(\sin\) | \(\cos\) | \(\tan\) |
|---|---|---|---|---|
| I | \(\theta\) | + | + | + |
| II | \(180° – \theta\) | + | − | − |
| III | \(\theta – 180°\) | − | − | + |
| IV | \(360° – \theta\) | − | + | − |
Memory trick: All Students Take Calculus → All, Sin, Tan, Cos (positive per quadrant I→IV)
✏️ Example: Find the reference angle for 310°
- 310° is in Quadrant IV
- Reference angle \(= 360° – 310° =\) 50°
6. The Six Trig Functions
Beyond sin, cos, and tan, there are three reciprocal functions:
| Function | Definition | Reciprocal of |
|---|---|---|
| \(\sin\theta\) | \(y/r\) | — |
| \(\cos\theta\) | \(x/r\) | — |
| \(\tan\theta\) | \(y/x\) | — |
| \(\csc\theta\) | \(r/y\) | \(\sin\theta\) |
| \(\sec\theta\) | \(r/x\) | \(\cos\theta\) |
| \(\cot\theta\) | \(x/y\) | \(\tan\theta\) |
On the unit circle, \(r = 1\), so: \(\sin\theta = y\) and \(\cos\theta = x\)
✏️ Example: Find all 6 trig functions for \(\theta = \dfrac{5\pi}{6}\)
- \(\dfrac{5\pi}{6}\) is in Quadrant II, reference angle \(= \dfrac{\pi}{6}\)
- \(\sin\!\left(\dfrac{5\pi}{6}\right) = \dfrac{1}{2}\) | \(\csc\!\left(\dfrac{5\pi}{6}\right) = 2\)
- \(\cos\!\left(\dfrac{5\pi}{6}\right) = -\dfrac{\sqrt{3}}{2}\) | \(\sec\!\left(\dfrac{5\pi}{6}\right) = -\dfrac{2\sqrt{3}}{3}\)
- \(\tan\!\left(\dfrac{5\pi}{6}\right) = -\dfrac{\sqrt{3}}{3}\) | \(\cot\!\left(\dfrac{5\pi}{6}\right) = -\sqrt{3}\)
7. Points on the Unit Circle
Given a point \((x, y)\) on the unit circle, you can find all 6 trig values directly:
- \(\sin\theta = y\)
- \(\cos\theta = x\)
- \(\tan\theta = y/x\)
- \(\csc\theta = 1/y\)
- \(\sec\theta = 1/x\)
- \(\cot\theta = x/y\)
✏️ Example: Point \(\left(-\dfrac{\sqrt{3}}{2},\ \dfrac{1}{2}\right)\) is on the unit circle. Find all 6 trig values.
- \(\sin\theta = \dfrac{1}{2}\)
- \(\cos\theta = -\dfrac{\sqrt{3}}{2}\)
- \(\tan\theta = \dfrac{1/2}{-\sqrt{3}/2} = -\dfrac{1}{\sqrt{3}} = -\dfrac{\sqrt{3}}{3}\)
- \(\csc\theta = 2\)
- \(\sec\theta = -\dfrac{2}{\sqrt{3}} = -\dfrac{2\sqrt{3}}{3}\)
- \(\cot\theta = -\sqrt{3}\)
8. Arc Length & Sector Area
These formulas require the angle to be in radians.
| Formula | Variables |
|---|---|
| Arc Length: \(s = r\theta\) | \(r\) = radius, \(\theta\) = angle in radians |
| Sector Area: \(A = \frac{1}{2}r^2\theta\) | \(r\) = radius, \(\theta\) = angle in radians |
✏️ Example 1: A circle has radius 6 and \(\theta = \dfrac{\pi}{3}\). Find the arc length.
$$s = r\theta = 6 \times \frac{\pi}{3} = 2\pi$$✏️ Example 2: Same circle. Find the sector area.
$$A = \frac{1}{2}r^2\theta = \frac{1}{2} \times 36 \times \frac{\pi}{3} = 6\pi$$✏️ Example 3: Find arc length for \(r = 10\), \(\theta = \dfrac{3\pi}{4}\)
$$s = 10 \times \frac{3\pi}{4} = \frac{15\pi}{2}$$