Trigonometry

8.

Applications of Trigonometry

8.1.

8.1.1.

Statement of Law of Sines

8.1.1.1.

a/sin A = b/sin B = c/sin C

8.1.1.2.

Alternative Forms

8.1.2.

Proof of Law of Sines

8.1.2.1.

8.1.2.2.

Geometric Derivation

8.1.3.

Solving AAS Triangles

8.1.3.1.

Angle-Angle-Side Cases

8.1.3.2.

Finding Unknown Angle

8.1.3.3.

Finding Unknown Sides

8.1.4.

Solving ASA Triangles

8.1.4.1.

Angle-Side-Angle Cases

8.1.4.2.

Solution Strategy

8.1.5.

The Ambiguous Case (SSA)

8.1.5.1.

Side-Side-Angle Configuration

8.1.5.2.

Determining Number of Solutions

8.1.5.2.1.

No Solution Cases

8.1.5.2.2.

One Solution Cases

8.1.5.2.3.

Two Solution Cases

8.1.5.3.

Solving for All Possible Triangles

8.1.5.4.

Geometric Interpretation

8.2.

8.2.1.

Statement of Law of Cosines

8.2.1.1.

c² = a² + b² - 2ab cos C

8.2.1.2.

Alternative Forms for Other Sides

8.2.2.

Proof of Law of Cosines

8.2.2.1.

Coordinate Geometry Method

8.2.2.2.

8.2.3.

Solving SAS Triangles

8.2.3.1.

Side-Angle-Side Cases

8.2.3.2.

Finding Unknown Side

8.2.3.3.

Finding Remaining Angles

8.2.4.

Solving SSS Triangles

8.2.4.1.

Side-Side-Side Cases

8.2.4.2.

Finding All Angles

8.2.4.3.

Verification Methods

8.2.5.

Relationship to Pythagorean Theorem

8.2.5.1.

Special Case When C = 90°

8.2.5.2.

Generalization Concept

8.3.

Area of Triangles

8.3.1.

Area Using Sine

8.3.1.1.

Formula: Area = ½ab sin C

8.3.1.2.

Derivation from Basic Area Formula

8.3.1.3.

Applications to Oblique Triangles

8.3.2.

Heron's Formula

8.3.2.1.

Statement: Area = √[s(s-a)(s-b)(s-c)]

8.3.2.2.

Semi-perimeter Definition

8.3.2.3.

Derivation Process

8.3.2.4.

When to Use Heron's Formula

8.3.2.5.

Computational Considerations

8.3.3.

Choosing Appropriate Area Formula

8.3.3.1.

Given Information Analysis

8.3.3.2.

Efficiency Considerations

8.4.

Introduction to Vectors

8.4.1.

Definition of Vector

8.4.1.1.

Magnitude and Direction

8.4.1.2.

Geometric Representation

8.4.1.3.

Distinction from Scalars

8.4.2.

Vector Notation

8.4.2.1.

8.4.2.2.

Magnitude Notation

8.4.2.3.

Unit Vector Notation

8.4.3.

Vector Operations

8.4.3.1.

Vector Addition

8.4.3.1.1.

Parallelogram Method

8.4.3.1.2.

Component Method

8.4.3.1.3.

Properties of Addition

8.4.3.2.

Vector Subtraction

8.4.3.2.1.

Geometric Interpretation

8.4.3.2.2.

Component Method

8.4.3.3.

Scalar Multiplication

8.4.3.3.1.

Effect on Magnitude

8.4.3.3.2.

Effect on Direction

8.4.4.

8.4.4.1.

Definition: a⃗ · b⃗ = |a⃗||b⃗|cos θ

8.4.4.2.

Component Formula

8.4.4.3.

Properties of Dot Product

8.4.4.3.1.

Commutative Property

8.4.4.3.2.

Distributive Property

8.4.4.4.

Applications of Dot Product

8.4.5.

Angle Between Vectors

8.4.5.1.

Formula Using Dot Product

8.4.5.2.

Calculating Angles

8.4.5.3.

Perpendicular Vectors

8.4.5.4.

Parallel Vectors

8.5.

Polar Coordinates

8.5.1.

Polar Coordinate System

8.5.1.1.

Origin (Pole) and Polar Axis

8.5.1.2.

Distance and Angle Coordinates

8.5.1.3.

Notation (r, θ)

8.5.2.

Plotting Points in Polar Coordinates

8.5.2.1.

Positive and Negative r Values

8.5.2.2.

Angle Measurement Conventions

8.5.2.3.

Multiple Representations

8.5.3.

Converting Between Coordinate Systems

8.5.3.1.

Polar to Rectangular

8.5.3.1.1.

8.5.3.1.2.

8.5.3.2.

Rectangular to Polar

8.5.3.2.1.

r = √(x² + y²)

8.5.3.2.2.

θ = arctan(y/x) with quadrant adjustment

8.5.4.

Graphing Polar Equations

8.5.4.1.

Basic Polar Curves

8.5.4.1.1.

8.5.4.1.2.

Lines Through Origin: θ = α

8.5.4.1.3.

Cardioids: r = a(1 + cos θ)

8.5.4.1.4.

Rose Curves: r = a cos(nθ)

8.5.4.1.5.

Limaçons: r = a + b cos θ

8.5.4.2.

8.5.4.2.1.

Polar Axis Symmetry

8.5.4.2.2.

Line θ = π/2 Symmetry

8.5.4.2.3.

Origin Symmetry

8.5.4.3.

Graphing Strategies

8.5.4.3.1.

Table of Values Method

8.5.4.3.2.

Symmetry Utilization

8.6.

Complex Numbers in Trigonometric Form

8.6.1.

8.6.1.1.

Real Axis and Imaginary Axis

8.6.1.2.

Plotting Complex Numbers

8.6.1.3.

Geometric Interpretation

8.6.2.

Polar Form of Complex Numbers

8.6.2.1.

Modulus (Absolute Value)

8.6.2.1.1.

r = |z| = √(a² + b²)

8.6.2.2.

Argument (Angle)

8.6.2.2.1.

8.6.2.2.2.

Principal Argument

8.6.2.3.

Trigonometric Form

8.6.2.3.1.

z = r(cos θ + i sin θ)

8.6.2.3.2.

Euler's Notation: z = re^(iθ)

8.6.3.

Converting Between Forms

8.6.3.1.

Rectangular to Polar Form

8.6.3.2.

Polar to Rectangular Form

8.6.3.3.

Handling Different Quadrants

8.6.4.

Operations in Polar Form

8.6.4.1.

Multiplication of Complex Numbers

8.6.4.1.1.

Multiply Moduli, Add Arguments

8.6.4.1.2.

(r₁∠θ₁)(r₂∠θ₂) = r₁r₂∠(θ₁ + θ₂)

8.6.4.2.

Division of Complex Numbers

8.6.4.2.1.

Divide Moduli, Subtract Arguments

8.6.4.2.2.

(r₁∠θ₁)/(r₂∠θ₂) = (r₁/r₂)∠(θ₁ - θ₂)

8.6.5.

De Moivre's Theorem

8.6.5.1.

Statement: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)

8.6.5.2.

8.6.5.3.

Applications to Powers

8.6.5.3.1.

Computing High Powers

8.6.5.3.2.

Simplifying Calculations

8.6.6.

Roots of Complex Numbers

8.6.6.1.

nth Roots Formula

8.6.6.1.1.

ⁿ√r[cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)]

8.6.6.2.

Finding All nth Roots

8.6.6.2.1.

k = 0, 1, 2, ..., n-1

8.6.6.3.

Geometric Interpretation

8.6.6.3.1.

Regular n-gon Vertices

8.6.6.3.2.

Symmetry Properties

8.6.6.4.

8.6.6.4.1.

Special Case: nth Roots of 1

8.6.6.4.2.

Applications in Algebra

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7. Solving Trigonometric Equations

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1. Foundations of Trigonometry