Chapter 16: Trigonometric Functions

Overview

Trigonometric functions extend basic trigonometric ratios to functions of real numbers, providing powerful tools for modeling periodic phenomena. This chapter explores the graphs of trigonometric functions, fundamental trigonometric identities, addition formulas, and their applications in solving equations and modeling real-world situations. Mastery of trigonometric functions is essential for understanding periodic motion, waves, oscillations, and many applications in physics and engineering.

Learning Objectives

After completing this chapter, you will be able to:

• Sketch and analyze graphs of trigonometric functions
• Apply trigonometric identities to simplify expressions
• Use addition formulas to evaluate trigonometric values
• Solve trigonometric equations
• Apply trigonometric functions to model periodic phenomena

Key Concepts

16.1 Trigonometric Functions and Their Graphs

Basic Trigonometric Functions

The six basic trigonometric functions are:

• Sine: sin(x)
• Cosine: cos(x)
• Tangent: tan(x) = sin(x)/cos(x)
• Cosecant: csc(x) = 1/sin(x)
• Secant: sec(x) = 1/cos(x)
• Cotangent: cot(x) = 1/tan(x) = cos(x)/sin(x)

Properties of Sine and Cosine Functions

Sine Function (y = sin(x)):

• Period: 2π
• Amplitude: 1 (maximum value)
• Range: [-1, 1]
• Zeros: x = nπ, n ∈ ℤ
• Maximum points: x = π/2 + 2nπ
• Minimum points: x = 3π/2 + 2nπ

Cosine Function (y = cos(x)):

• Period: 2π
• Amplitude: 1
• Range: [-1, 1]
• Zeros: x = π/2 + nπ, n ∈ ℤ
• Maximum points: x = 2nπ
• Minimum points: x = π + 2nπ

Tangent Function

Tangent Function (y = tan(x)):

• Period: π
• Range: (-∞, ∞)
• Undefined: x = π/2 + nπ, n ∈ ℤ
• Zeros: x = nπ, n ∈ ℤ

16.2 Trigonometric Identities

Fundamental Identities

Pythagorean Identities:

$$\sin^2 x + \cos^2 x = 1$$ $$\tan^2 x + 1 = \sec^2 x$$ $$1 + \cot^2 x = \csc^2 x$$

Reciprocal Identities:

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

Quotient Identities:

$$\tan x = \frac{\sin x}{\cos x}, \quad \cot x = \frac{\cos x}{\sin x}$$

Co-Function Identities

$$\sin\left(\frac{\pi}{2} - x\right) = \cos x, \quad \cos\left(\frac{\pi}{2} - x\right) = \sin x$$ $$\tan\left(\frac{\pi}{2} - x\right) = \cot x$$

Even-Odd Identities

Even Functions:

$$\cos(-x) = \cos x, \quad \sec(-x) = \sec x$$

Odd Functions:

$$\sin(-x) = -\sin x, \quad \tan(-x) = -\tan x, \quad \csc(-x) = -\csc x, \quad \cot(-x) = -\cot x$$

16.3 Addition Formulas

Sine and Cosine Addition Formulas

Sine Addition:

$$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$$

Cosine Addition:

$$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$$

Tangent Addition:

$$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$$

Double Angle Formulas

Sine Double Angle:

$$\sin(2A) = 2 \sin A \cos A$$

Cosine Double Angle:

$$\cos(2A) = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$$

Tangent Double Angle:

$$\tan(2A) = \frac{2 \tan A}{1 - \tan^2 A}$$

Half Angle Formulas

$$\sin\left(\frac{A}{2}\right) = \pm\sqrt{\frac{1 - \cos A}{2}}$$ $$\cos\left(\frac{A}{2}\right) = \pm\sqrt{\frac{1 + \cos A}{2}}$$ $$\tan\left(\frac{A}{2}\right) = \frac{1 - \cos A}{\sin A} = \frac{\sin A}{1 + \cos A}$$

Important Formulas and Methods

Key Trigonometric Formulas

Graph Transformation Methods

For y = a sin(bx + c) + d:

• |a|: Amplitude (vertical stretch)
• b: Period = 2π/|b| (horizontal stretch/compression)
• c/b: Phase shift = -c/b (horizontal shift)
• d: Vertical shift

For y = a cos(bx + c) + d:

• Same parameters as sine, but different phase

Solved Examples

Example 1: Basic Trigonometric Evaluations

Evaluate the following: a) sin(π/3) b) cos(π/4) c) tan(π/6) d) sin(3π/2)

Solution:

a) sin(π/3) = √3/2

b) cos(π/4) = √2/2

c) tan(π/6) = sin(π/6)/cos(π/6) = (1/2)/(√3/2) = 1/√3 = √3/3

d) sin(3π/2) = -1

Example 2: Simplifying Using Identities

Simplify the following expressions: a) si$n^2$x + co$s^2$x + ta$n^2$x se$c^2$x b) (1 + ta$n^2$x)(1 - si$n^2$x) c) sin x cos x / tan x

Solution:

a) si$n^2$x + co$s^2$x + ta$n^2$x se$c^2$x = 1 + (si$n^2$x/co$s^2$x)(1/co$s^2$x) = 1 + si$n^2$x/co$s^4$x

b) (1 + ta$n^2$x)(1 - si$n^2$x) = se$c^2$x × co$s^2$x = (1/co$s^2$x) × co$s^2$x = 1

c) sin x cos x / tan x = sin x cos x / (sin x/cos x) = sin x cos x × (cos x/sin x) = co$s^2$x

Example 3: Addition Formula Applications

Find the exact values: a) sin(75°) b) cos(105°) c) tan(15°)

Solution:

a) sin(75°) = sin(45° + 30°) = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = √2/2 (√3/2 + 1/2) = √2(√3 + 1)/4

b) cos(105°) = cos(60° + 45°) = cos60°cos45° - sin60°sin45° = (1/2)(√2/2) - (√3/2)(√2/2) = √2/4 - √6/4 = (√2 - √6)/4

c) tan(15°) = tan(45° - 30°) = (tan45° - tan30°)/(1 + tan45°tan30°) = (1 - 1/√3)/(1 + 1 × 1/√3) = (√3 - 1)/(√3 + 1) Multiply numerator and denominator by (√3 - 1): = (3 - 2√3 + 1)/(3 - 1) = (4 - 2√3)/2 = 2 - √3

Example 4: Double Angle Applications

Find sin(2x) if sin x = 3/5 and x is in quadrant II.

Solution:

Since sin x = 3/5 and x is in quadrant II: cos x = -√(1 - si$n^2$x) = -√(1 - 9/25) = -√(16/25) = -4/5

sin(2x) = 2 sin x cos x = 2 × (3/5) × (-4/5) = 2 × (-12/25) = -24/25

Example 5: Solving Trigonometric Equations

Solve for x in [0, 2π]: a) 2 sin x + 1 = 0 b) cos(2x) = cos x c) 2 co$s^2$x - 3 cos x + 1 = 0

Solution:

a) 2 sin x + 1 = 0 ⇒ sin x = -1/2 Solutions: x = 7π/6, 11π/6

b) cos(2x) = cos x Using double angle: 2co$s^2$x - 1 = cos x 2co$s^2$x - cos x - 1 = 0 (2cos x + 1)(cos x - 1) = 0 cos x = -1/2 or cos x = 1 Solutions: x = 2π/3, 4π/3, 0, 2π

c) 2 co$s^2$x - 3 cos x + 1 = 0 Let u = cos x: 2$u^2$ - 3u + 1 = 0 (2u - 1)(u - 1) = 0 cos x = 1/2 or cos x = 1 Solutions: x = π/3, 5π/3, 0, 2π

Example 6: Graph Transformations

Sketch the graph of y = 2 sin(3x - π/2) + 1 for one period.

Solution:

For y = 2 sin(3x - π/2) + 1:

• Amplitude = 2
• Period = 2π/3
• Phase shift = (π/2)/3 = π/6
• Vertical shift = 1

Transformations:

1. Start with y = sin x
2. Vertical stretch by factor 2: y = 2 sin x
3. Horizontal compression by factor 3: y = 2 sin(3x)
4. Horizontal shift right by π/6: y = 2 sin(3(x - π/6))
5. Vertical shift up by 1: y = 2 sin(3(x - π/6)) + 1

Key points in one period [0, 2π/3]:

• x = π/6, y = 1 (midpoint, shifted)
• x = π/6 + π/6 = π/3, y = 1 + 2 = 3 (max)
• x = π/6 + 2π/6 = π/2, y = 1 (zero)
• x = π/6 + 3π/6 = 2π/3, y = 1 - 2 = -1 (min)

Example 7: Complex Trigonometric Simplification

Simplify: (sin 4x + sin 2x)/(cos 4x - cos 2x)

Solution:

Using sum-to-product formulas: sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2) cos A - cos B = -2 sin((A+B)/2) sin((A-B)/2)

Numerator: sin 4x + sin 2x = 2 sin(3x) cos(x) Denominator: cos 4x - cos 2x = -2 sin(3x) sin(x)

Result: [2 sin(3x) cos(x)] / [-2 sin(3x) sin(x)] = -cos(x)/sin(x) = -cot(x)

Example 8: Trigonometric Equation with Multiple Solutions

Solve: 3 co$s^2$x - si$n^2$x = 2 for all real x.

Solution:

Use co$s^2$x = 1 - si$n^2$x: 3(1 - si$n^2$x) - si$n^2$x = 2 3 - 3si$n^2$x - si$n^2$x = 2 3 - 4si$n^2$x = 2 -4si$n^2$x = -1 si$n^2$x = 1/4 sin x = ±1/2

Solutions: x = π/6, 5π/6, 7π/6, 11π/6 + 2nπ, n ∈ ℤ

Mathematical Derivations

Derivation of Cosine Addition Formula

Using the distance formula and unit circle: Points: (cos A, sin A) and (cos B, sin B) Distance between them is the same whether measured directly or through rotation by angle A-B.

Distance formula: √[(cos A - cos B)² + (sin A - sin B)²]

Rotation formula: √[2 - 2 cos(A-B)]

Setting equal and squaring both sides: (cos A - cos B)² + (sin A - sin B)² = 2 - 2 cos(A-B)

Expanding and simplifying leads to: cos(A-B) = cos A cos B + sin A sin B

Derivation of Double Angle Formulas

Set B = A in addition formulas: cos(2A) = cos A cos A - sin A sin A = co$s^2$A - si$n^2$A sin(2A) = sin A cos A + cos A sin A = 2 sin A cos A

Derivation of Half Angle Formulas

Start with double angle formula: cos(2A) = 2co$s^2$A - 1 Let θ = 2A, so A = θ/2 cos θ = 2co$s^2$(θ/2) - 1 co$s^2$(θ/2) = (1 + cos θ)/2

Similarly from cos θ = 1 - 2si$n^2$(θ/2): si$n^2$(θ/2) = (1 - cos θ)/2

Real-World Applications

1. Physics and Engineering

Simple Harmonic Motion:

• Position: x(t) = A cos(ωt + φ)
• Velocity: v(t) = -Aω sin(ωt + φ)
• Acceleration: a(t) = -Aω² cos(ωt + φ)

Example: A mass-spring system with amplitude 5 cm, frequency 2 Hz, and phase 0. Find position at t = 0.1 s.

2. Architecture and Construction

Structural Analysis:

• Truss calculations using trigonometry
• Roof pitch angles
• Load distribution

Example: Calculate the angle of a roof pitch if the rise is 3 feet and run is 8 feet.

3. Electrical Engineering

AC Circuits:

• Voltage: V(t) = $V_0$ sin(ωt)
• Current: I(t) = $I_0$ sin(ωt + φ)
• Power calculations

Example: Find the phase difference between voltage and current in an RLC circuit.

4. Music and Acoustics

Sound Waves:

• Frequency and amplitude relationships
• Wave interference patterns
• Musical intervals

Example: Two musical notes with frequencies 440 Hz and 445 Hz produce beats. Find the beat frequency.

Complex Problem-Solving Techniques

Problem: Prove the identity: (sin 3x + sin x)/(cos 3x + cos x) = tan 2x

Solution:

Using sum-to-product formulas: Numerator: sin 3x + sin x = 2 sin(2x) cos(x) Denominator: cos 3x + cos x = 2 cos(2x) cos(x)

Result: [2 sin(2x) cos(x)] / [2 cos(2x) cos(x)] = sin(2x)/cos(2x) = tan(2x) ✓

Problem: Find all solutions to sin(2x) + √2 cos(2x) = 0

Solution:

Divide both sides by cos(2x) (assuming cos(2x) ≠ 0): tan(2x) = -√2

2x = arctan(-√2) + nπ x = ½ arctan(-√2) + nπ/2, n ∈ ℤ

Check where cos(2x) = 0: 2x = π/2 + nπ ⇒ x = π/4 + nπ/2 For these values, sin(2x) = ±1, so the equation becomes ±1 = 0, which is false.

Final solution: x = ½ arctan(-√2) + nπ/2, n ∈ ℤ

Problem: Express 3 sin x - 4 cos x in the form R sin(x - α)

Solution:

We want 3 sin x - 4 cos x = R sin(x - α) = R(sin x cos α - cos x sin α)

Equate coefficients: R cos α = 3 R sin α = 4

Square and add: $R^2$ = 3² + 4² = 25 ⇒ R = 5 tan α = 4/3 ⇒ α = arctan(4/3)

Therefore: 3 sin x - 4 cos x = 5 sin(x - arctan(4/3))

Summary Points

• Trigonometric Functions: sin, cos, tan, csc, sec, cot with periodic properties
• Graphs: Sine and cosine have period 2π, amplitude, and range [-1,1]
• Identities: Fundamental relationships used to simplify expressions
• Addition Formulas: Enable calculation of compound angles
• Double/Half Angles: Special cases for multiple and half angles
• Applications: Periodic phenomena, waves, oscillations in physics and engineering

Common Mistakes to Avoid

1. Identity errors - Memorize and verify basic identities before use
2. Domain issues - Watch for undefined values (tan x at π/2, etc.)
3. Solution sets - Include all solutions within specified intervals
4. Angle units - Be careful with degrees vs radians in calculations
5. Sign errors - Pay attention to signs based on quadrants

SPM Exam Tips

Exam Strategies

1. Memorize formulas - Know all basic identities and addition formulas
2. Show working - Step-by-step simplification for identity proofs
3. Sketch graphs - Understand transformations for graph problems
4. Check solutions - Verify solutions satisfy original equations
5. Use multiple methods - Try different approaches when stuck

Key Exam Topics

• Trigonometric identities (25% of questions)
• Addition and double angle formulas (25% of questions)
• Graph sketching and transformations (20% of questions)
• Trigonometric equations (25% of questions)
• Applications (5% of questions)

Time Management Tips

• Basic identities: 3-4 minutes
• Formula applications: 4-5 minutes
• Equation solving: 5-6 minutes
• Graph problems: 6-7 minutes
• Complex proofs: 8-10 minutes

Practice Problems

Level 1: Basic Evaluations

1. Evaluate: a) sin(π/6) b) cos(π/3) c) tan(π/4) d) csc(π/2)

2. Find the exact values: a) sin(45° + 30°) b) cos(60° - 45°) c) tan(75°)

Level 2: Identities and Simplification

3. Simplify: a) si$n^2$x + co$s^2$x - ta$n^2$x se$c^2$x b) (1 + ta$n^2$x) co$s^2$x c) sin x / cos x + cos x / sin x

4. Prove the identity: (1 - co$s^2$x) / si$n^2$x = 1

Level 3: Equations

5. Solve for x in [0, 2π]: a) 2 sin x = √3 b) cos(2x) = 1/2 c) ta$n^2$x - 3 = 0

6. Solve: 2 co$s^2$x + 3 cos x + 1 = 0

Level 4: Graphs and Transformations

7. Sketch one period of: a) y = 2 sin x + 1 b) y = 3 cos(2x) c) y = tan(x/2)

8. For y = 2 cos(3x - π/4) + 2, find: a) Amplitude b) Period c) Phase shift d) Vertical shift

Level 5: Applications

9. Physics: A mass-spring system oscillates with amplitude 4 cm and period 2π seconds. Find the equation of motion if it starts at maximum displacement.

10. Engineering: An AC voltage is given by V(t) = 120 sin(120πt). Find the frequency and the voltage at t = 0.001 s.

Did You Know? 📚

Trigonometry originated from Greek astronomers who needed to calculate positions of stars and planets. The word "trigonometry" comes from Greek words meaning "triangle measurement." The sine function was originally defined as the length of a chord in a circle, not the ratio we use today. Trigonometric identities were systematically developed by Indian and Islamic mathematicians before being adopted in Europe. Today, trigonometry is fundamental in fields from navigation and architecture to quantum mechanics and signal processing.

Quick Reference Guide

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Trigonometric functions provide the mathematical language for describing periodic phenomena. Master these concepts to understand waves, oscillations, and countless applications in science, engineering, and mathematics.