IB Mathematics AA – Topic 3: Geometry & Trigonometry

Comprehensive Guide to Trigonometric Functions

Introduction to Trigonometric Functions

Trigonometric functions extend beyond simple right-triangle ratios to become powerful periodic functions that model cyclical phenomena throughout mathematics, physics, and engineering. From sound waves and light oscillations to planetary motion and alternating current, these functions describe any repeating pattern in nature and technology.

Key concepts: The unit circle provides a geometric foundation for defining sine, cosine, and tangent for all angles, not just acute ones. These circular functions produce beautiful wave-like graphs with specific properties: period, amplitude, and phase shift. Understanding these characteristics enables modeling of real-world periodic behavior.

Why this matters: Trigonometric identities—equations true for all angle values—allow simplification of complex expressions and proofs. Solving trigonometric equations requires understanding periodicity and using inverse functions. These skills are essential for calculus, Fourier analysis, signal processing, and countless applications in STEM fields.

In this guide: We'll master the unit circle with exact values, explore how circular functions produce periodic graphs, learn essential trigonometric identities (Pythagorean, double angle, compound angle), develop systematic techniques for solving trigonometric equations, and analyze transformations of trigonometric graphs—all critical for IB exam success.

1. The Unit Circle and Trigonometric Ratios

Definition Using the Unit Circle

Unit Circle Definition

A circle with radius 1 centered at the origin

For a point \((x, y)\) on the unit circle at angle \(\theta\):

\(\cos\theta = x\)

\(\sin\theta = y\)

\(\tan\theta = \frac{y}{x} = \frac{\sin\theta}{\cos\theta}\)

Angle \(\theta\) measured counterclockwise from positive x-axis

Exact Values - Essential Angles

Key Angles to Memorize:

Angle (degrees)	Angle (radians)	\(\sin\theta\)	\(\cos\theta\)	\(\tan\theta\)
0°	0	0	1	0
30°	\(\frac{\pi}{6}\)	\(\frac{1}{2}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{\sqrt{3}}\)
45°	\(\frac{\pi}{4}\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{\sqrt{2}}{2}\)	1
60°	\(\frac{\pi}{3}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{2}\)	\(\sqrt{3}\)
90°	\(\frac{\pi}{2}\)	1	0	undefined

CAST Diagram - Signs in Four Quadrants

CAST Rule (which ratios are positive):

Quadrant I (0° to 90°): All positive (sine, cosine, tangent)
Quadrant II (90° to 180°): Sine positive only
Quadrant III (180° to 270°): Tangent positive only
Quadrant IV (270° to 360°): Cosine positive only

Memory aid: "All Students Take Calculus" (counterclockwise from Quadrant I)

⚠ Common Pitfalls:

Radians vs degrees: Always check calculator mode!
Reference angles: Use CAST to determine sign, then use reference angle for magnitude
Exact values: Memorize the key angles—they appear frequently on IB exams
Domain restrictions: tan undefined at odd multiples of \(\frac{\pi}{2}\)

2. Trigonometric Identities

Pythagorean Identities

Fundamental Pythagorean Identity

\(\sin^2\theta + \cos^2\theta = 1\)

Derived from unit circle: \(x^2 + y^2 = 1\)

Related Identities:

\(1 + \tan^2\theta = \sec^2\theta\)

\(1 + \cot^2\theta = \csc^2\theta\)

Double Angle Identities

Double Angle Formulas:

Sine: \(\sin(2\theta) = 2\sin\theta\cos\theta\)

Cosine (three forms):

\(\cos(2\theta) = \cos^2\theta - \sin^2\theta\)

\(\cos(2\theta) = 2\cos^2\theta - 1\)

\(\cos(2\theta) = 1 - 2\sin^2\theta\)

Tangent: \(\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}\)

Compound Angle Identities

Addition and Subtraction Formulas:

Sine:

\(\sin(A + B) = \sin A\cos B + \cos A\sin B\)

\(\sin(A - B) = \sin A\cos B - \cos A\sin B\)

Cosine:

\(\cos(A + B) = \cos A\cos B - \sin A\sin B\)

\(\cos(A - B) = \cos A\cos B + \sin A\sin B\)

Tangent:

\(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}\)

\(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A\tan B}\)

💡 Identity Tips:

Start with Pythagorean identity—most versatile for simplification
Double angle formulas come from compound angle formulas (set A = B)
Choose the form that matches what you're given or need to find
Practice proving identities by working from more complex to simpler side

Example 1: Using Trigonometric Identities

Problem: Prove the identity: \(\frac{1 - \cos(2\theta)}{\sin(2\theta)} = \tan\theta\)

Solution:

Start with left-hand side (LHS):

\(\text{LHS} = \frac{1 - \cos(2\theta)}{\sin(2\theta)}\)

Step 1: Use double angle formulas

\(\cos(2\theta) = 1 - 2\sin^2\theta\) (choose this form)

\(\sin(2\theta) = 2\sin\theta\cos\theta\)

Step 2: Substitute

\(= \frac{1 - (1 - 2\sin^2\theta)}{2\sin\theta\cos\theta}\)

\(= \frac{1 - 1 + 2\sin^2\theta}{2\sin\theta\cos\theta}\)

\(= \frac{2\sin^2\theta}{2\sin\theta\cos\theta}\)

Step 3: Simplify

\(= \frac{\sin\theta}{\cos\theta}\)

\(= \tan\theta = \text{RHS}\)

Identity proven ✓

3. Trigonometric Graphs and Circular Functions

Basic Trigonometric Graphs

Properties of Basic Functions:

\(y = \sin x\)

Period: \(2\pi\) (or 360°)
Amplitude: 1 (maximum value)
Domain: all real numbers
Range: \([-1, 1]\)
Starts at origin \((0, 0)\)

\(y = \cos x\)

Period: \(2\pi\) (or 360°)
Amplitude: 1
Domain: all real numbers
Range: \([-1, 1]\)
Starts at maximum \((0, 1)\)
Same as \(\sin x\) shifted left by \(\frac{\pi}{2}\)

\(y = \tan x\)

Period: \(\pi\) (or 180°) - shorter period!
No amplitude (unbounded)
Domain: \(x \neq \frac{\pi}{2} + n\pi\) (odd multiples of \(\frac{\pi}{2}\))
Range: all real numbers
Vertical asymptotes at \(x = \frac{\pi}{2}, \frac{3\pi}{2}, ...\)

Transformations of Trigonometric Graphs

General Transformed Form

\(f(x) = a\sin(b(x - c)) + d\)

(Same structure for cosine and tangent)

Parameters:

\(a\): Amplitude (vertical stretch) - distance from center to max/min
\(b\): Affects period - new period = \(\frac{2\pi}{b}\) for sin/cos, \(\frac{\pi}{b}\) for tan
\(c\): Horizontal shift (phase shift) - positive \(c\) shifts right
\(d\): Vertical shift - moves the center line up/down

⚠ Graph Pitfalls:

Period formula: Period = \(\frac{2\pi}{|b|}\) for sin/cos, not \(2\pi b\)
Amplitude only for sin/cos: Tangent has no amplitude (unbounded)
Phase shift direction: \(+c\) inside shifts right, \(-c\) shifts left
Range after transformation: With amplitude \(a\) and shift \(d\), range is \([d-a, d+a]\)

4. Solving Trigonometric Equations

General Strategy

Systematic Approach:

Isolate the trigonometric function (if possible)
Find the principal solution using inverse trig functions
Use periodicity to find all solutions in the given interval
Apply CAST rule to find solutions in other quadrants
Check solutions in the original equation

Common Equation Types

Standard Forms:

Type 1: Simple equations

\(\sin x = k\), \(\cos x = k\), \(\tan x = k\)

Use inverse functions and CAST rule

Type 2: Quadratic form

\(a\sin^2 x + b\sin x + c = 0\)

Let \(u = \sin x\), solve quadratic, then find \(x\)

Type 3: Using identities

\(\sin(2x) = \cos x\)

Use identities to express in single function, then solve

Type 4: Factoring

\(\sin x\cos x = \sin x\)

Factor: \(\sin x(\cos x - 1) = 0\), solve each factor

💡 Solving Tips:

Always note the domain: \(0 \leq x \leq 2\pi\) or \(0° \leq x \leq 360°\)
Principal value is the reference angle in first quadrant
For \(\sin x = k\): solutions at \(x\) and \(\pi - x\) (symmetric about \(\frac{\pi}{2}\))
For \(\cos x = k\): solutions at \(x\) and \(2\pi - x\) (symmetric about x-axis)
For \(\tan x = k\): solutions separated by \(\pi\) (period of tangent)

Example 2: Solving Trigonometric Equations (IB-Style)

Problem: Solve \(2\sin^2 x + \sin x - 1 = 0\) for \(0 \leq x \leq 2\pi\)

Solution:

Step 1: Recognize as quadratic in \(\sin x\)

Let \(u = \sin x\)

Equation becomes: \(2u^2 + u - 1 = 0\)

Step 2: Factor the quadratic

\(2u^2 + u - 1 = (2u - 1)(u + 1) = 0\)

Step 3: Solve for \(u\)

\(2u - 1 = 0\) → \(u = \frac{1}{2}\)

or \(u + 1 = 0\) → \(u = -1\)

Step 4: Solve \(\sin x = \frac{1}{2}\)

Principal value: \(x = \sin^{-1}(\frac{1}{2}) = \frac{\pi}{6}\)

Using CAST: sine positive in Quadrants I and II

Second solution: \(x = \pi - \frac{\pi}{6} = \frac{5\pi}{6}\)

Step 5: Solve \(\sin x = -1\)

\(x = \frac{3\pi}{2}\) (sine equals -1 at bottom of unit circle)

Complete solution: \(x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2}\)

Or in degrees: \(x = 30°, 150°, 270°\)

📋 Trigonometric Functions Quick Reference

Concept	Formula/Rule	Notes
Pythagorean	\(\sin^2\theta + \cos^2\theta = 1\)	Most fundamental identity
Double Angle (sin)	\(\sin(2\theta) = 2\sin\theta\cos\theta\)	For doubling angles
Period (sin/cos)	\(\frac{2\pi}{b}\) for \(f(bx)\)	Period of \(2\pi\) when \(b=1\)
Period (tan)	\(\frac{\pi}{b}\) for \(\tan(bx)\)	Half the period of sin/cos
Amplitude	\(\|a\|\) in \(a\sin(bx)\)	Distance from center to max

🎯 IB Exam Strategy

Common Question Types:

"Find exact values": Use unit circle and special angles
"Prove the identity": Work from one side to the other using identities
"Solve for \(0 \leq x \leq 2\pi\)": Find all solutions using CAST and periodicity
"State period and amplitude": From \(a\sin(bx)\), period = \(\frac{2\pi}{b}\), amplitude = \(|a|\)
"Sketch the graph": Identify transformations, mark key points

Key Reminders:

Memorize exact values for 0°, 30°, 45°, 60°, 90° and their radian equivalents
Check calculator mode: radians vs degrees
Use CAST to find all solutions in the given interval
Pythagorean identity is your best friend for simplification
Show all steps when proving identities

🎉 Master Trigonometric Functions!

Trigonometric functions are the mathematical language of waves, oscillations, and periodic phenomena. From sound and light to electrical signals and planetary orbits, these functions describe the rhythms of the universe. Mastering the unit circle, identities, graphs, and equation-solving techniques prepares you for advanced calculus and countless applications!

Key Success Factors:

✓ Unit circle: \(\cos\theta = x\), \(\sin\theta = y\), memorize special angles
✓ Pythagorean identity: \(\sin^2\theta + \cos^2\theta = 1\) (most used!)
✓ Period of \(\sin(bx)\) and \(\cos(bx)\) is \(\frac{2\pi}{b}\)
✓ Period of \(\tan(bx)\) is \(\frac{\pi}{b}\) (half of sin/cos)
✓ Use CAST rule to find all solutions in given interval
✓ Double angle formulas from compound angles (set A = B)

Master the Circle • Use Identities • Find All Solutions

Master trigonometric functions and excel in IB Mathematics! 🚀