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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 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

15.4 Double-Angle and Half-Angle Identities

Setting both angles equal in the angle addition identities gives the double-angle identities, and solving those for the smaller angle gives the half-angle identities.

Double angles

Problem 15.4.1. Find and . Test your identities with and .

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and Testing with : and For : , and

Problem 15.4.2. Find in terms of .

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Half angles

Problem 15.4.3. Use the double-angle identities for sine and cosine with to find and in terms of . Test with .

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From we get . And so If is an angle in quadrant I, it is positive; for cosine we can say it is negative (by “it” I mean our ) iff is in quadrant II or III. So OR Using we get similar ().

Problem 15.4.4. it’s weird.

Problem 15.4.5. Show that .

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Well, If instead idk why we drop ?

Using double and half angles

Example 15.4.6. Prove that by using three well-chosen triangles on the Cartesian plane.

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Problem 15.4.7. Compute the number of degrees in the smallest positive angle that satisfies the equation .

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(I think?). Wait, no: pull out . So and So and , , so .

Problem 15.4.8. Find .

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In radians this is , with and .

Using and :

Instead: , so .

Exercises

Exercise 15.4.9.

Exercise 15.4.10. If , then what is ?

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Exercise 15.4.11. Compute .

Exercise 15.4.12. Use the angle addition and double-angle identities with and .

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Exercise 15.4.13. Prove that for all for which is defined.

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Multiply top and bottom by :

Exercise 15.4.14. Show that .

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Exercise 15.4.15. If and , then prove that

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Exercise 15.4.16. If and , then compute .

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, and so

Exercise 15.4.17. Solve the equation for .

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Squaring: so , that is, Check:

Exercise 15.4.18. (a) Prove that .

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(b) Do the same for .

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(c) Suppose for some angle , with . Determine .

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so

Let’s start again: Using : (and the other way round, ). So so , and