Chapter 16: Ratios and Graphs of Trigonometric Functions

Overview

Welcome to Chapter 16 of Form 5 Mathematics! This chapter introduces you to the fascinating world of trigonometry and trigonometric functions. You'll learn about trigonometric ratios, the unit circle, and the graphs of sine, cosine, and tangent functions. These concepts are essential for understanding periodic phenomena, waves, oscillations, and have applications in physics, engineering, and many other fields.

What You'll Learn:

• Understand trigonometric ratios for angles 0° to 360°
• Use the unit circle to determine trigonometric values
• Sketch and analyze graphs of trigonometric functions
• Understand the effects of parameter changes on trigonometric graphs

Learning Objectives

After completing this chapter, you will be able to:

• Make and verify conjectures about sine, cosine, and tangent values of angles in quadrants II, III, and IV using corresponding reference angles
• Solve problems involving trigonometric ratios
• Sketch graphs of trigonometric functions
• Study and make generalizations about the effects of changing constants a, b, c in y = a sin(bx) + c, y = a cos(bx) + c, and y = a tan(bx) + c
• Solve problems involving these graphs

Unit Circle and Trigonometric Identities

Unit Circle Visualization

Basic Trigonometric Identities

Pythagorean Identity:

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

Reciprocal Identities:

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

Quotient Identity:

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

Co-function Identities:

$$\sin(90° - \theta) = \cos \theta, \quad \cos(90° - \theta) = \sin \theta$$

Angle Sum and Difference Identities

Sum Identities:

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

Difference Identities:

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

Trigonometric Graph Analysis

Basic Function Graphs

Parameter Effects on Trigonometric Graphs

Amplitude Changes (a parameter)

Examples:

• y = 2 sin(x): Amplitude = 2 (stretched vertically)
• y = 0.5 cos(x): Amplitude = 0.5 (compressed vertically)
• y = -3 sin(x): Amplitude = 3, reflected over x-axis

Period Changes (b parameter)

Period Formulas:

• Sine/Cosine: Period = $\frac{360°}{|b|}$
• Tangent: Period = $\frac{180°}{|b|}$

Examples:

• y = sin(2x): Period = 180° (compressed horizontally)
• y = cos(0.5x): Period = 720° (stretched horizontally)
• y = tan(-3x): Period = 60°, reflected

Vertical Shift (c parameter)

Examples:

• y = sin(x) + 2: Shifted up by 2 units
• y = cos(x) - 1.5: Shifted down by 1.5 units

Combined Transformations

When multiple parameters are changed, apply transformations in this order:

1. Horizontal scaling (b parameter)
2. Horizontal shifting (phase shift)
3. Vertical scaling (a parameter)
4. Vertical shifting (c parameter)

Example: y = 2 sin(3x) + 1

Phase Shift and Horizontal Transformations

Phase Shift Formulas

For functions of the form:

• y = a sin(b(x - h)) + k
• y = a cos(b(x - h)) + k
• y = a tan(b(x - h)) + k

Phase Shift: h (horizontal shift)

• h > 0: Shift right
• h < 0: Shift left

Period: $\frac{360°}{|b|}$ (sine/cosine), $\frac{180°}{|b|}$ (tangent)

Phase Shift Examples

Example 1: y = 3 cos(2(x - 45°)) - 2

• Amplitude = 3
• Period = 360°/2 = 180°
• Phase shift = 45° right
• Vertical shift = 2 down

Example 2: y = 4 sin(0.5(x + 30°)) + 1

• Amplitude = 4
• Period = 360°/0.5 = 720°
• Phase shift = 30° left
• Vertical shift = 1 up

Special Angles and Exact Values

Common Angle Values

Angle Relationships

Trigonometric Equation Solving

Basic Equation Types

Type 1: Single Function

$$\sin \theta = k, \quad \cos \theta = k, \quad \tan \theta = k$$

Type 2: Multiple Angles

$$\sin(b\theta) = k, \quad \cos(b\theta) = k, \quad \tan(b\theta) = k$$

Type 3: Multiple Functions

$$a \sin \theta + b \cos \theta = c$$

Solving Strategies

Strategy 1: Reference Angle Method

Example: Solve 2 sin θ = √3 for 0° ≤ θ ≤ 360°

1. Divide by 2: sin θ = √3/2
2. Reference angle: θ = 60°
3. Solutions in QI and QII: θ = 60°, 120°

Strategy 2: Factor and Solve

Example: Solve sin(2θ) - sin θ = 0 for 0° ≤ θ ≤ 360°

1. Use double angle: 2 sin θ cos θ - sin θ = 0
2. Factor: sin θ(2 cos θ - 1) = 0
3. Solve: sin θ = 0 or cos θ = 1/2
4. Solutions: θ = 0°, 180°, 60°, 300°

Strategy 3: Use Identities

Example: Solve si$n^2$ θ + sin θ - 2 = 0 for 0° ≤ θ ≤ 360°

1. Let x = sin θ: $x^2$ + x - 2 = 0
2. Factor: (x + 2)(x - 1) = 0
3. Solve: x = -2 or x = 1
4. Only x = 1 valid: sin θ = 1
5. Solution: θ = 90°

Real-world Wave Applications

Simple Harmonic Motion

General Equation: y = A sin(ωt + φ) + D

Where:

• A = Amplitude
• ω = Angular frequency (2πf)
• t = Time
• φ = Phase angle
• D = Equilibrium position

Example: Spring-mass system

• Spring constant k, mass m
• Frequency f = $\frac{1}{2π}\sqrt{\frac{k}{m}}$
• Period T = $\frac{1}{f} = 2π\sqrt{\frac{m}{k}}$

Wave Applications

Engineering Applications

Resonance Frequency:

$$f_r = \frac{1}{2π}\sqrt{\frac{1}{LC}}$$

Impedance in AC Circuits:

$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$

Power in AC Circuits:

$$P = V_{rms} \cdot I_{rms} \cdot \cos \phi$$

Advanced Graph Analysis

Graph Transformations Summary

Composite Functions

Example: y = 3 sin(2(x - 30°)) + 1

1. Start with y = sin(x)
2. Horizontal compression by 2: y = sin(2x), period = 180°
3. Horizontal shift right 30°: y = sin(2(x - 30°))
4. Vertical stretch by 3: y = 3 sin(2(x - 30°)), amplitude = 3
5. Vertical shift up 1: y = 3 sin(2(x - 30°)) + 1

Graph Sketching Steps

1. Identify parameters: a, b, h, k
2. Calculate properties: amplitude, period, phase shift, vertical shift
3. Find key points: zeros, maxima, minima, asymptotes
4. Sketch one period: Mark key points and smooth curves
5. Repeat pattern: Extend graph periodically
6. Label axes: Show scale and important values

Trigonometric Ratios

For any angle θ in standard position:

$$\sin \theta = \frac{y}{r}, \quad \cos \theta = \frac{x}{r}, \quad \tan \theta = \frac{y}{x}$$

Where (x,y) is a point on the terminal side and r = √($x^2$ + $y^2$).

Signs in Different Quadrants

Reference Angle Method

To find trigonometric values for any angle θ:

1. Find reference angle α
2. Find trigonometric value for α
3. Apply correct sign based on quadrant

Example: sin 150°

• Reference angle: 180° - 150° = 30°
• sin 30° = 0.5
• Quadrant II: sin positive
• sin 150° = +0.5

Graph Parameters

For y = a sin(bx) + c, y = a cos(bx) + c, y = a tan(bx) + c:

• Amplitude: |a| (for sine and cosine)
• Period: 360°/|b| (for sine and cosine), 180°/|b| (for tangent)
• Vertical Shift: c
• Phase Shift: Additional horizontal shift if present

Key Concepts

Unit Circle

The unit circle is a circle with radius 1 unit centered at the origin. Points on the circle have coordinates (cos θ, sin θ) where θ is the angle from the positive x-axis.

Key Properties:

• Radius = 1
• Center at origin (0,0)
• θ measured counterclockwise from positive x-axis
• Coordinates: (cos θ, sin θ)

Quadrants

The Cartesian plane is divided into four quadrants:

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

Mnemonic: "All Students Take Calculus"

• All (Quadrant I): All positive
• Students (Quadrant II): Sine positive
• Take (Quadrant III): Tangent positive
• Calculus (Quadrant IV): Cosine positive

Reference Angle

The reference angle α is the acute angle made by the terminal side of angle θ with the x-axis.

Reference Angle by Quadrant:

• Quadrant I: α = θ
• Quadrant II: α = 180° - θ
• Quadrant III: α = θ - 180°
• Quadrant IV: α = 360° - θ

Trigonometric Graphs

Basic Graph Shapes:

• Sine: y = sin x - Waveform starting at origin
• Cosine: y = cos x - Waveform starting at maximum
• Tangent: y = tan x - Periodic with vertical asymptotes

Key Parameters:

• Amplitude: |a| - Maximum displacement from center line
• Period: 360°/|b| - Length of one complete cycle
• Vertical Shift: c - Up/down shift of entire graph

Important Formulas and Methods

Trigonometric Ratios

For any angle θ in standard position:

$$\sin \theta = \frac{y}{r}, \quad \cos \theta = \frac{x}{r}, \quad \tan \theta = \frac{y}{x}$$

Where (x,y) is a point on the terminal side and r = √($x^2$ + $y^2$).

Signs in Different Quadrants

Reference Angle Method

To find trigonometric values for any angle θ:

1. Find reference angle α
2. Find trigonometric value for α
3. Apply correct sign based on quadrant

Example: sin 150°

• Reference angle: 180° - 150° = 30°
• sin 30° = 0.5
• Quadrant II: sin positive
• sin 150° = +0.5

Graph Parameters

For y = a sin(bx) + c, y = a cos(bx) + c, y = a tan(bx) + c:

• Amplitude: |a| (for sine and cosine)
• Period: 360°/|b| (for sine and cosine), 180°/|b| (for tangent)
• Vertical Shift: c
• Phase Shift: Additional horizontal shift if present

Step-by-Step Solved Examples

Example 1: Trigonometric Values Using Reference Angles

Problem: Find the exact values: a) sin 135° b) cos 210° c) tan 315°

Solution: a) sin 135°:

• 135° in Quadrant II
• Reference angle: 180° - 135° = 45°
• sin 45° = √2/2
• Quadrant II: sin positive
• sin 135° = +√2/2

b) cos 210°:

• 210° in Quadrant III
• Reference angle: 210° - 180° = 30°
• cos 30° = √3/2
• Quadrant III: cos negative
• cos 210° = -√3/2

c) tan 315°:

• 315° in Quadrant IV
• Reference angle: 360° - 315° = 45°
• tan 45° = 1
• Quadrant IV: tan negative
• tan 315° = -1

Answer: a) √2/2, b) -√3/2, c) -1

Example 2: Graph Sketching - Sine Function

Problem: Sketch the graph of y = 2 sin(3x) + 1

Solution: Step 1: Identify parameters

• a = 2, b = 3, c = 1
• Amplitude = |2| = 2
• Period = 360°/|3| = 120°
• Vertical shift = 1
• Range: [1-2, 1+2] = [-1, 3]

Step 2: Key points for one period (0° to 120°):

• x = 0°: y = 2 sin(0) + 1 = 1
• x = 30°: y = 2 sin(90°) + 1 = 2(1) + 1 = 3
• x = 60°: y = 2 sin(180°) + 1 = 2(0) + 1 = 1
• x = 90°: y = 2 sin(270°) + 1 = 2(-1) + 1 = -1
• x = 120°: y = 2 sin(360°) + 1 = 2(0) + 1 = 1

Step 3: Plot and repeat pattern

Answer: Graph with amplitude 2, period 120°, shifted up by 1 unit

Example 3: Graph Sketching - Cosine Function

Problem: Sketch y = 3 cos(2x) - 2

Solution: Step 1: Identify parameters

• a = 3, b = 2, c = -2
• Amplitude = |3| = 3
• Period = 360°/|2| = 180°
• Vertical shift = -2
• Range: [-2-3, -2+3] = [-5, 1]

Step 2: Key points for one period (0° to 180°):

• x = 0°: y = 3 cos(0°) - 2 = 3(1) - 2 = 1
• x = 45°: y = 3 cos(90°) - 2 = 3(0) - 2 = -2
• x = 90°: y = 3 cos(180°) - 2 = 3(-1) - 2 = -5
• x = 135°: y = 3 cos(270°) - 2 = 3(0) - 2 = -2
• x = 180°: y = 3 cos(360°) - 2 = 3(1) - 2 = 1

Step 3: Plot and repeat pattern

Answer: Graph with amplitude 3, period 180°, shifted down by 2 units

Example 4: Tangent Function Analysis

Problem: Analyze y = 4 tan(0.5x)

Solution: Step 1: Identify parameters

• a = 4, b = 0.5, c = 0
• Amplitude: Not defined for tangent
• Period = 180°/|0.5| = 360°
• Vertical shift: 0
• Range: All real numbers

Step 2: Find asymptotes: tan(0.5x) undefined when 0.5x = 90° + 180°n x = 180° + 360°n

Step 3: Key points for one period:

• x = 0°: y = 4 tan(0°) = 0
• x = 90°: y = 4 tan(45°) = 4(1) = 4
• x = 180°: undefined (asymptote)
• x = 270°: y = 4 tan(135°) = 4(-1) = -4
• x = 360°: y = 4 tan(180°) = 0

Answer: Tangent graph with period 360°, vertical stretch by 4, asymptotes at x = 180° + 360°n

Example 5: Real-world Application - Wave Motion

Problem: A wave is described by y = 5 sin(2πt) where y is displacement in cm and t is time in seconds. Find: a) Amplitude and period b) Displacement at t = 0.25 seconds c) Time when displacement is maximum

Solution: a) Amplitude and Period: y = 5 sin(2πt)

• Amplitude = 5 cm
• Period = 2π/|2π| = 1 second

b) Displacement at t = 0.25: y = 5 sin(2π × 0.25) = 5 sin(π/2) = 5(1) = 5 cm

c) Maximum displacement: Maximum occurs when sin(2πt) = 1 2πt = π/2 + 2πn t = 0.25 + n seconds (where n is integer)

Answer: a) Amplitude 5 cm, period 1 second; b) 5 cm; c) t = 0.25 + n seconds

Real-world Applications

1. Physics and Engineering

• Wave Motion: Sound waves, light waves, water waves
• Simple Harmonic Motion: Pendulums, springs, oscillations
• AC Circuits: Voltage and current variations over time

2. Architecture and Construction

• Roof Design: Trigonometric calculations for angles
• Structural Analysis: Periodic loading calculations
• Vibrations: Building response to earthquakes and wind

3. Navigation and Surveying

• Position Calculations: GPS and coordinate systems
• Surveying: Land measurement and angle calculations
• Astronomy: Celestial navigation and position tracking

4. Music and Acoustics

• Sound Waves: Frequency and amplitude relationships
• Musical Notes: Trigonometric functions for sound waves
• Audio Processing: Signal analysis and filtering

Important Terms

Summary Points

• Unit Circle: Provides (cos θ, sin θ) coordinates for all angles
• Quadrant Signs: "All Students Take Calculus" for trig signs
• Reference Angles: Help find trig values for any angle
• Graph Parameters:
  • Amplitude: |a| - Maximum displacement
  • Period: 360°/|b| - One complete cycle
  • Vertical Shift: c - Up/down movement
• Sine: Starts at (0,0), smooth wave
• Cosine: Starts at maximum, smooth wave
• Tangent: Has asymptotes, periodic with period 180°/|b|

Practice Tips for SPM Students

1. Master Reference Angles

• Practice finding reference angles for all quadrants
• Memorize trig values for common angles (0°, 30°, 45°, 60°, 90°)
• Learn to apply correct signs based on quadrants

2. Graph Sketching

• Practice sketching basic sine, cosine, and tangent graphs
• Understand parameter effects (a, b, c) on graphs
• Learn to identify key points and asymptotes

3. Real-world Applications

• Relate trigonometric functions to wave motion
• Practice physics and engineering applications
• Understand periodic phenomena in everyday life

4. Common Mistakes to Avoid

• Confusing period calculations for different functions
• Incorrect sign application in different quadrants
• Misidentifying amplitude and vertical shift
• Forgetting tangent has undefined values

SPM Exam Tips

Paper 1 (Multiple Choice)

• Look for angle quadrant relationships
• Remember reference angle methods
• Practice quick graph parameter identification
• Use elimination method for difficult questions

Paper 2 (Structured)

• Show all reference angle calculations
• Demonstrate graph sketching steps
• Explain parameter effects clearly
• Use proper mathematical notation throughout

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Did You Know? Trigonometry originated from ancient civilizations' need for astronomy and navigation. The word "trigonometry" comes from Greek words meaning "triangle measurement." Today, it's essential for everything from GPS navigation to music production and computer graphics!

Next Chapter: In Chapter 7, you'll explore measures of dispersion for grouped data, building on your statistics knowledge from Form 4 but applying it to categorized data.

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• Chapter 5: Congruence, Enlargement and Transformations
• Chapter 7: Measures of Dispersion for Grouped Data