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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

15.2 Basic Trigonometric Identities

The identities on this page come straight from the definitions on the unit circle. The problems use them to rewrite expressions, to solve equations, and to relate the functions at complementary and supplementary angles.

Theorem 15.2.1 (Most important trigonometric identities).

and

Pythagorean identities

Problem 15.2.2. (a) Write in terms of and .

(b) Solve for in terms of .

(c) Use to find .

(d) Find all with that satisfy the equation.

Show solution

(a)

(b)

(c) So

(d) So , , are options. But wait! Since and , we need to plug into the original equation.

So for , , and are undefined. So we have an extraneous/bogus solution.

Next, : , , , so So is a valid solution.

Finally, : which is invalid! So our only solution is .

Problem 15.2.3. Prove the identity .

Show solution

which was our given identity.

Problem 15.2.4. Show that .

Show solution

By the previous problem we know , so which is exactly the definition of .

Cofunction and reflection identities

Problem 15.2.5. Represent the following in terms of , , (and too).

(a) (b) (c)

Show solution

(a) , and So . For : undefined?

(b) .

(c) So .

Problem 15.2.6. Continue.

Show solution

(a)

(b)

(c)

(d)

So and .

Problem 15.2.7. Find the smallest possible angle such that .

Show solution

Well, Also , so :

Exercises

Exercise 15.2.8. Prove .

Show solution

Exercise 15.2.9. Prove that .

Show solution

Since and , we have .

Exercise 15.2.10. Use the identity proved above to solve (the first problem on this page) without converting into sines and cosines.

Show solution

So we have and . Squaring the first, Now , so so .

(In red:) idk where I’m going wrong, I’m tired.

Exercise 15.2.11. Find in terms of or .

Show solution

Exercise 15.2.12. Find the value of the product

Show solution

The middle three are and the outer two, , cancel out. Maybe a better way than brute force computation: Ah, so they cancel out since by the identity. So do and , and , which has of . TLDR: the product is .

Exercise 15.2.13. Find the range of where is not an integer multiple of .

Show solution

Since , the cancels: Multiplying through by :

A detour, writing everything with : which didn’t lead anywhere.

Back to the bracket: Here and , so this is Not sure, seems premature to say.