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 and Proof
1. 7.1 Sets: Notation, Subsets and Operations
2. 7.2 Intuitive Proofs: Dominoes on a Chessboard
8 Equations, Functions and Graphs
1. 8.1 Solving Equations and Inequalities
2. 8.2 Functions: Definition, Notation and Domain
3. 8.3 Properties of Functions from Their Graphs
4. 8.4 Piecewise Functions, Basic Graphs and Transformations
5. 8.5 Special Classes of Functions
6. 8.6 More Piecewise-Defined Functions
7. 8.7 Functional Equations
9 Quadratics
10 Polynomials
11 Sequences, Series and Induction
1. 11.1 Arithmetic Sequences
2. 11.2 Arithmetic Series
3. 11.3 Geometric Sequences
4. 11.4 Mathematical Induction
5. 11.5 More Sequences and Series
12 Exponents, Logarithms and Radicals
1. 12.1 Exponents and Logarithms
2. 12.2 Radicals
13 Identities and Inequalities
1. 13.1 Identities and Manipulations
2. 13.2 Inequalities
3. 13.3 More Inequalities
4. 13.4 Some Advanced Strategies
14 Graph Theory, Combinatorics and Probability
1. 14.1 Graph Theory
2. 14.2 Combinatorics
3. 14.3 Probability and Counting
Integrated Maths II
15 Trigonometry
16 Parametric Equations, Polar Coordinates and Complex Numbers
17 The Real Numbers
18 Groups, Rings and Fields
1. 18.1 Integral Domains: Definition and Elementary Properties
19 Vector Spaces and Topology
1. 19.1 Vector Spaces
2. 19.2 Topology
Integrated Maths III and IV
20 Integrated Maths III
1. 20.1 Advanced Linear Algebra
2. 20.2 Advanced Group Theory
3. 20.3 Real Analysis
4. 20.4 Topology
21 Integrated Maths IV
1. 21.1 Complex Analysis
2. 21.2 Differential Equations
3. 21.3 Lie Groups
4. 21.4 Geometry
5. 21.5 Differential Geometry

notes / Integrated Mathematics / chapter 15: Trigonometry

15.6 Solving Triangles with the Laws of Sines and Cosines

The law of cosines and the law of sines find the missing sides and angles of a triangle from the ones we know.

The law of cosines

Theorem 15.6.1 (Law of cosines). where is the angle between sides and , opposite side (for example, sides and with between them).

Solving right triangles

Example 15.6.2. (a) In , we have , , and . Find and .

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and . Angle, side, angle: and .

(b) In , , , and . Find .

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, so . Then , and is .

Exercises

The law of cosines

Exercise 15.6.3. Find if .

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Exercise 15.6.4. .

The law of sines

Exercise 15.6.5. A triangle has angles , and , with side opposite the angle. Find the side opposite .

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with .

Exercise 15.6.6. Find if .

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

Exercise 15.6.7. Find if .

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