📐 Trigonometry Basics

Master the unit circle, special angles, and right triangle relationships. These fundamentals are essential for all AMC trigonometry problems.

🎯 Key Ideas

Unit Circle: Circle with radius 1 centered at origin. Coordinates $(x,y)$ correspond to $(\cos\theta, \sin\theta)$.

SOH-CAH-TOA: For right triangles: $\sin = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan = \frac{\text{opposite}}{\text{adjacent}}$.

Special Angles: $30°$, $45°$, $60°$ (and their radian equivalents) have exact values that must be memorized.

⭕ Unit Circle

Basic Setup

• Center: $(0,0)$
• Radius: $1$
• Angle: Measured counterclockwise from positive $x$-axis
• Coordinates: $(\cos\theta, \sin\theta)$

Quadrant Signs

📊 Special Angles

Exact Values (Memorize These!)

Pro Move: Memory Tricks

• 30-60-90 triangle: Sides in ratio $1 : \sqrt{3} : 2$
• 45-45-90 triangle: Sides in ratio $1 : 1 : \sqrt{2}$
• Sine values: $\frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}$ (increasing)
• Cosine values: $\frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{1}{2}$ (decreasing)

🔺 Right Triangle Relationships

SOH-CAH-TOA

For right triangle with angle $\theta$:

• $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

Pythagorean Identity

• $\sin^2\theta + \cos^2\theta = 1$
• $\tan\theta = \frac{\sin\theta}{\cos\theta}$

Example: Find exact values of all trig functions for $30°$

Solution: Using 30-60-90 triangle with sides $1, \sqrt{3}, 2$:

• $\sin 30° = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2}$
• $\cos 30° = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}$
• $\tan 30° = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

🔄 Reference Angles

Definition: The acute angle between the terminal side and the $x$-axis.

Finding Reference Angles

• Quadrant I: Reference angle = angle
• Quadrant II: Reference angle = $180° - \text{angle}$
• Quadrant III: Reference angle = angle - $180°$
• Quadrant IV: Reference angle = $360° - \text{angle}$

Using Reference Angles

1. Find the reference angle
2. Determine the quadrant
3. Apply appropriate signs

Example: Find $\sin 150°$

Solution:

1. Reference angle: $180° - 150° = 30°$
2. Quadrant: II (sine is positive)
3. Value: $\sin 150° = \sin 30° = \frac{1}{2}$

📐 Cofunctions

Cofunction Identities:

• $\sin(90° - \theta) = \cos\theta$
• $\cos(90° - \theta) = \sin\theta$
• $\tan(90° - \theta) = \cot\theta$

Example: Find $\cos 60°$ using cofunctions

Solution:

• $\cos 60° = \sin(90° - 60°) = \sin 30° = \frac{1}{2}$

🎯 AMC-Style Worked Example

Problem: Find the exact value of $\sin 75°$.

Solution:

1. Use angle addition: $75° = 45° + 30°$
2. Apply sine addition formula: $\sin(45° + 30°) = \sin 45° \cos 30° + \cos 45° \sin 30°$
3. Substitute known values:
   • $\sin 45° = \frac{\sqrt{2}}{2}$, $\cos 30° = \frac{\sqrt{3}}{2}$
   • $\cos 45° = \frac{\sqrt{2}}{2}$, $\sin 30° = \frac{1}{2}$
4. Calculate: $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}$

Answer: $\frac{\sqrt{6} + \sqrt{2}}{4}$

🔍 Common Traps & Fixes

Trap: Wrong quadrant signs

Fix: Always determine the quadrant first, then apply the correct signs.

Trap: Confusing degrees and radians

Fix: Be consistent with units. Convert if necessary: $180° = \pi$ radians.

Trap: Forgetting reference angles

Fix: For angles outside $[0°, 90°]$, always find the reference angle first.

Trap: Undefined tangent values

Fix: $\tan\theta$ is undefined when $\cos\theta = 0$ (at $90°$ and $270°$).

📋 Quick Reference

Unit Circle Values

• $(1,0)$ → $\cos 0° = 1$, $\sin 0° = 0$
• $(\frac{\sqrt{3}}{2}, \frac{1}{2})$ → $\cos 30° = \frac{\sqrt{3}}{2}$, $\sin 30° = \frac{1}{2}$
• $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ → $\cos 45° = \frac{\sqrt{2}}{2}$, $\sin 45° = \frac{\sqrt{2}}{2}$
• $(\frac{1}{2}, \frac{\sqrt{3}}{2})$ → $\cos 60° = \frac{1}{2}$, $\sin 60° = \frac{\sqrt{3}}{2}$
• $(0,1)$ → $\cos 90° = 0$, $\sin 90° = 1$

Essential Identities

• $\sin^2\theta + \cos^2\theta = 1$
• $\tan\theta = \frac{\sin\theta}{\cos\theta}$
• $\sin(90° - \theta) = \cos\theta$
• $\cos(90° - \theta) = \sin\theta$

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