Skip to content

Integrated Mathematics

Basic Mathematics
1 Arithmetic as Structure
1. 1.1 Why Arithmetic Has Rules
2. 1.2 Sets: The Language of Structure
3. 1.3 Properties of Addition
4. 1.4 Properties of Multiplication
5. 1.5 Negation and Additive Inverses
6. 1.6 Subtraction and Integer Arithmetic
7. 1.7 Reciprocals and Multiplicative Inverses
8. 1.8 Division and Fractions
9. 1.9 Exponents: Repeated Multiplication
10. 1.10 Zero and Negative Exponents
11. 1.11 Exploring Structure: Symmetry, Clocks, and Composition
12. 1.12 Order of Operations and Structure Summary
2 Algebraic Foundations
1. 2.1 What is a Variable?
2. 2.2 Combining Like Terms
3. 2.3 Distribution and Factoring
4. 2.4 What is an Equation?
5. 2.5 Solving Multi-Step Linear Equations
6. 2.6 Special Cases: No Solution and Infinite Solutions
7. 2.7 Word Problems I: Translating to Equations
8. 2.8 Word Problems II: Geometry and Rate Problems
9. 2.9 Inequalities
10. 2.10 Polynomials and Like Terms
11. 2.11 Polynomial Multiplication and Factoring I
12. 2.12 Factoring II and Part II Summary
3 Number Theory and Patterns
1. 3.1 Divisibility Rules and Tests
2. 3.2 Primes and Composite Numbers
3. 3.3 Prime Factorization and Fundamental Theorem
4. 3.4 GCD and LCM
5. 3.5 The Euclidean Algorithm
6. 3.6 Introduction to Modular Arithmetic
7. 3.7 Computing with Congruences
8. 3.8 Linear Congruences and Diophantine Equations
9. 3.9 Patterns in Number Theory
10. 3.10 Part III Summary: Number Theory as Structure
4 Geometric Reasoning
1. 4.1 Points, Lines, and Planes: The Undefined Terms
2. 4.2 Angles: Measuring and Classifying
3. 4.3 Parallel Lines and Transversals
4. 4.4 Triangles: Classification and Angle Sum
5. 4.5 Congruent Triangles
6. 4.6 Proof Techniques in Geometry
7. 4.7 Quadrilaterals and Their Properties
8. 4.8 Perimeter and Area: Rectangles, Triangles, Parallelograms
9. 4.9 The Pythagorean Theorem
10. 4.10 Similar Triangles
11. 4.11 Circles: Definitions and Basic Properties
12. 4.12 Part IV Summary: Geometry as Logical Structure
5 Counting, Probability and Combinatorics
1. 5.1 Counting Principles: Addition and Multiplication
2. 5.2 Permutations
3. 5.3 Combinations
4. 5.4 Pascal’s Triangle and Binomial Theorem
5. 5.5 Counting with Restrictions and Casework
6. 5.6 Introduction to Probability
7. 5.7 Compound Events: AND, OR, Conditional
8. 5.8 Expected Value
9. 5.9 Geometric Probability and Continuous Outcomes
10. 5.10 Part V Summary: Counting as Structure
6 Functions, Graphs and Synthesis
1. 6.1 What is a Function?
2. 6.2 Linear Functions and Their Graphs
3. 6.3 Systems of Linear Equations
4. 6.4 Quadratic Functions and Parabolas
5. 6.5 Introduction to Coordinate Geometry
6. 6.6 Transformations of Functions
7. 6.7 Function Composition and Inverses
8. 6.8 Course Summary: Mathematics as Structure
Integrated Maths I
7 Sets
1. 7.1 Sets
8 College Algebra
9 Intermediate Algebra (Art of Problem Solving)
1. 1 Basic Techniques for Solving Equations
2. 2 Functions Review
3. 3 Complex Numbers
4. 4 Quadratics
5. 5 Conics
6. 6 Polynomial Division
7. 7 Polynomial Roots Part I
8. 8 Polynomial Roots Part II
9. 9 Factoring Multivariable Polynomials
10. 10 Sequences and Series
11. 11 Identities, Manipulations, and Induction
12. 12 Inequalities
13. 13 Exponents and Logarithms
14. 14 Radicals
15. 15 Special Classes of Functions
16. 16 Piecewise Defined Functions
17. 17 More Sequences and Series
18. 18 More Inequalities
19. 19 Functional Equations
20. 20 Some Advanced Strategies
21. · Other Problems
10 Proofs (Cummings)
1. 1 Intuitive Proofs
11 Graph Theory, Combinatorics and Probability
1. 11.1 Graph Theory
2. 11.2 Combinatorics
3. 11.3 Probability and Counting
Integrated Maths II
12 Precalculus (Art of Problem Solving)
13 Calculus (Apostol, Volume I)
14 Groups, Rings and Fields (Birkhoff and Mac Lane)
1. 1.1 Definition of an Integral Domain
2. 1.2 Elementary Properties of Integral Domains
15 Vector Spaces and Topology
1. 15.1 Vector Spaces
2. 15.2 Topology
Integrated Maths III and IV
16 Integrated Maths III
1. 16.1 Advanced Linear Algebra
2. 16.2 Advanced Group Theory
3. 16.3 Real Analysis
4. 16.4 Topology
17 Integrated Maths IV
1. 17.1 Complex Analysis
2. 17.2 Differential Equations
3. 17.3 Lie Groups
4. 17.4 Geometry
5. 17.5 Differential Geometry

notes / Integrated Mathematics / chapter 12: Precalculus (Art of Problem Solving)

2a Practice: Trigonometric Equations and Triangles

Sinusoids and algebra

For :

Example 2a.1. Solve .

Show solution

Then and , giving the points and . Check: and .

Example 2a.2. Solve .

Show solution

Setting this equal to and squaring: Cancelling : so or .

Law of cosines

Theorem 2a.3 (Law of cosines). where is the angle between sides and , opposite side (for example, sides and with between them).

Example 2a.4. Find if .

Show solution

Next: .

Law of sines

Example 2a.5. A triangle has angles , and , with side opposite the angle. Find the side opposite .

Show solution

with .

Example 2a.6. Find if .

Show solution

A sketch of the sine curve meets the height twice: both angles work.

Next: .

Solving trigonometric equations

Note. Rule of note: when arc… (it is at a peak), it will only have one solution (per period).

Example 2a.7. Solve .

Show solution

Since , there are two families:

Example 2a.8. Solve .

Show solution

Example 2a.9. Solve .

Show solution

Example 2a.10. Solve .

Show solution

Example 2a.11. Solve cm.

Show solution

Angle addition and double angles

Theorem 2a.12.

With the values and :

Since :