Algebra 2.1

Chapter 2 · 2.1  |  Trigonometry Foundations

Chapter 2 · Trigonometry

2.1 Trigonometry —
Foundations

§ 5 Topics · Angles · Unit Circle · 6 Trig Functions · ASTC · Special Angles

Contents

1. Angle Measurement
2. The Unit Circle
3. Six Trigonometric Functions
4. Signs by Quadrant — ASTC Rule
5. Special Angle Values

01

Angle Measurement

There are two ways to measure angles in mathematics.

Degrees

The system used in high school geometry. A full circle = 360°.

Radians

The system used in advanced mathematics. A full circle = $2\pi$ radians.

1 radian is the angle where the arc length equals the radius.

          arc length = r 
          ╭───────────╮ 
         /             \ 
        /    1 radian   \ 
       │       •─────────│ ← radius =r 
        \                / 
         \              / 
          ╰────────────╯
          

Converting Between Degrees and Radians

$$180° = \pi \text{ radians}$$

Conversion	Formula
Degrees → Radians	Multiply by $\dfrac{\pi}{180}$
Radians → Degrees	Multiply by $\dfrac{180}{\pi}$

Common Angle Conversions

Degrees	Radians
$0°$	$0$
$30°$	$\dfrac{\pi}{6}$
$45°$	$\dfrac{\pi}{4}$
$60°$	$\dfrac{\pi}{3}$
$90°$	$\dfrac{\pi}{2}$
$180°$	$\pi$
$270°$	$\dfrac{3\pi}{2}$
$360°$	$2\pi$

💡

In university mathematics, radians are always assumed unless degrees are explicitly stated. Get comfortable with radians as early as possible.

02

The Unit Circle

The unit circle is a circle centered at the origin with radius = 1:

$$x^2 + y^2 = 1$$

For any angle $\theta$, the point where the ray from the origin intersects the unit circle directly defines $\cos\theta$ and $\sin\theta$:

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

💡

Memory tip: You only need to memorize the first quadrant values — the rest follow from symmetry.

Key Points on the Unit Circle

Angle $\theta$	Point $(x, y)$	$\cos\theta$	$\sin\theta$
$0$	$(1, 0)$	$1$	$0$
$\dfrac{\pi}{6}$	$\left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right)$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{2}$
$\dfrac{\pi}{4}$	$\left(\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2}\right)$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{2}}{2}$
$\dfrac{\pi}{3}$	$\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$	$\dfrac{1}{2}$	$\dfrac{\sqrt{3}}{2}$
$\dfrac{\pi}{2}$	$(0, 1)$	$0$	$1$
$\pi$	$(-1, 0)$	$-1$	$0$
$\dfrac{3\pi}{2}$	$(0, -1)$	$0$	$-1$
$2\pi$	$(1, 0)$	$1$	$0$

03

The Six Trigonometric Functions

Starting from a right triangle with angle $\theta$:

               /| 
              / | 
         hyp /  | opposite 
            /   | 
           /θ   | 
          /     |
         / ─────| 
         adjacent
         

Function	Abbreviation	Definition	Unit Circle
Sine	$\sin\theta$	$\dfrac{\text{opposite}}{\text{hypotenuse}}$	$y$
Cosine	$\cos\theta$	$\dfrac{\text{adjacent}}{\text{hypotenuse}}$	$x$
Tangent	$\tan\theta$	$\dfrac{\text{opposite}}{\text{adjacent}}$	$\dfrac{y}{x}$
Cosecant	$\csc\theta$	$\dfrac{\text{hypotenuse}}{\text{opposite}}$	$\dfrac{1}{y}$
Secant	$\sec\theta$	$\dfrac{\text{hypotenuse}}{\text{adjacent}}$	$\dfrac{1}{x}$
Cotangent	$\cot\theta$	$\dfrac{\text{adjacent}}{\text{opposite}}$	$\dfrac{x}{y}$

The last three are simply reciprocals of the first three:

$$\csc\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}$$

💡

Memory tip — SOH-CAH-TOA:

SOH

Sine = Opposite / Hypotenuse

CAH

Cosine = Adjacent / Hypotenuse

TOA

Tangent = Opposite / Adjacent

04

Signs by Quadrant — ASTC Rule

As $\theta$ moves through different quadrants, the signs of trig functions change.

Q2SSine only

Q1AAll positive

Q3TTangent only

Q4CCosine only

💡

Memory tip: All Students Take Calculus — reading counterclockwise from Q1.

Quadrant	Positive Functions	Reason (from unit circle)
Q1	All — $\sin, \cos, \tan$ positive	$x > 0$ and $y > 0$
Q2	Sine only — $\sin > 0$	$x < 0$ but $y > 0$
Q3	Tangent only — $\tan > 0$	$x < 0$ and $y < 0$, so $y/x > 0$
Q4	Cosine only — $\cos > 0$	$x > 0$ but $y < 0$

05

Special Angle Values

These are the values you must know by heart:

$\theta$	$\sin\theta$	$\cos\theta$	$\tan\theta$
$0$	$0$	$1$	$0$
$\dfrac{\pi}{6}$ $(30°)$	$\dfrac{1}{2}$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{\sqrt{3}}$
$\dfrac{\pi}{4}$ $(45°)$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{2}}{2}$	$1$
$\dfrac{\pi}{3}$ $(60°)$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{2}$	$\sqrt{3}$
$\dfrac{\pi}{2}$ $(90°)$	$1$	$0$	undefined

⚠️

$\tan\dfrac{\pi}{2}$ is undefined because $\cos\dfrac{\pi}{2} = 0$, and division by zero is not allowed.

Memory Trick — The $\sqrt{\phantom{x}}$ Pattern

$\sin$ values follow a neat square root pattern from $0°$ to $90°$:

$$\sin 0° = \frac{\sqrt{0}}{2} \quad \sin 30° = \frac{\sqrt{1}}{2} \quad \sin 45° = \frac{\sqrt{2}}{2} \quad \sin 60° = \frac{\sqrt{3}}{2} \quad \sin 90° = \frac{\sqrt{4}}{2}$$

And $\cos$ goes in reverse order:

$$\cos 0° = \frac{\sqrt{4}}{2} \quad \cos 30° = \frac{\sqrt{3}}{2} \quad \cos 45° = \frac{\sqrt{2}}{2} \quad \cos 60° = \frac{\sqrt{1}}{2} \quad \cos 90° = \frac{\sqrt{0}}{2}$$

✓

Summary Checklist

• ✓ Convert angles freely between degrees and radians
• ✓ Understand the unit circle and what $\cos\theta$ and $\sin\theta$ represent geometrically
• ✓ Know all 6 trig functions and their definitions
• ✓ Know the reciprocal relationships: $\csc$, $\sec$, $\cot$
• ✓ Determine the sign of any trig function in any quadrant using ASTC
• ✓ Recall all special angle values from memory