bythreedu · Math Tracks · Curriculum v1.0 · 2026

Angles, identities,
and the mathematics of
circular motion.

A complete trigonometry course covering the unit circle, all six trig functions, graphing, identities, equations, and triangle applications. Built for students who need trig as a foundation for precalculus, physics, or engineering — and who want to understand it, not just pass through it.

Duration~18weeks
Total Lessons18at your pace
Units6progressive phases
Assignments6+problem sets
DestinationPrecalc Ready
00Approach
Course Philosophy

The Unit Circle First

Most students treat trig as a collection of formulas. This course starts with the unit circle and derives everything from it — functions, graphs, identities, and inverse functions. The formulas become obvious once you understand where they come from.

Identities as Tools, Not Puzzles

Trig identities are usually taught as things to verify on a test. In this course they're tools — for simplifying expressions, solving equations, and building new formulas. You'll use them that way from the start.

01Curriculum
Learning Path
01
Unit 1 · Weeks 1–3
Angles, Radians, and the Unit Circle
3 lessons · Angle measure · Unit circle · Trig functions defined
radian measureunit circlesix trig functions
+
L1
Angles and Radian Measure
Degrees and radians as two measurement systems for the same thing. Arc length and sector area. Converting fluently in both directions without a formula sheet.
L2
Building the Unit Circle
Derived from 30-60-90 and 45-45-90 triangles — not memorized. Every coordinate understood geometrically before it is used.
L3
The Six Trig Functions
Sin, cos, tan and their reciprocals defined on the unit circle. Reference angles. Quadrant signs derived from the definitions, not memorized as ASTC.
Assignment 1
Evaluate 25 trig expressions using a unit circle you reconstructed from scratch. No reference sheet.
You own the unit circle and can evaluate any trig expression from it.
02
Unit 2 · Weeks 4–6
Trig Graphs
3 lessons · Sin and cos graphs · Other trig graphs · Transformations
amplitudeperiodphase shift
+
L4
Graphs of Sin and Cos
Amplitude, period, phase shift, and vertical shift as parameters — each one's effect understood independently before combining them.
L5
Graphs of Tan, Cot, Sec, Csc
Asymptotes, period, and shape for the four remaining functions. Connecting each asymptote back to a zero of the denominator function.
L6
Transformations of Trig Graphs
Reading any transformed trig function from its equation. Writing the equation from a graph. Identifying all parameters correctly.
Assignment 2
Sketch 8 trig functions from equations — no graphing calculator. Then write the equations for 4 graphs given key features.
You read the graph of any trig function from its equation.
03
Unit 3 · Weeks 7–9
Trig Identities
3 lessons · Fundamental identities · Sum and difference · Double and half angle
Pythagorean identitiessum formulasdouble angle
+
L7
Fundamental and Pythagorean Identities
The three Pythagorean identities derived from the unit circle. Quotient and reciprocal identities. Using identities to simplify expressions and verify equations.
L8
Sum and Difference Formulas
Where the sum formulas come from. Using them to find exact trig values for non-standard angles. Applying to simplify expressions.
L9
Double and Half Angle Formulas
Derived from the sum formulas — not memorized separately. Choosing which form of the double angle formula is most useful in context.
Assignment 3
Verify 5 trig identities and simplify 8 expressions — each step labeled with the identity used. No calculator.
You simplify and verify any trig identity without a reference sheet.
04
Unit 4 · Weeks 10–12
Solving Trig Equations
3 lessons · Basic trig equations · Equations requiring identities · General solutions
general solutionprincipal valueidentity substitution
+
L10
Solving Basic Trig Equations
Finding all solutions on [0, 2π) and in general. The difference between a principal value and all solutions. Why the unit circle is the right tool.
L11
Equations Requiring Identities
Using identities to convert equations into a solvable form. Factoring trig equations. Checking for extraneous solutions introduced by squaring.
L12
Multiple Angle Equations
Solving equations with 2θ, 3θ, or θ/2 — the substitution, the solution set, and converting back. Not losing solutions by dividing improperly.
Assignment 4
Solve 12 trig equations — 4 basic, 4 requiring identity substitution, 4 multiple angle. Write the general solution for each.
You solve any trig equation and write the complete solution set.
05
Unit 5 · Weeks 13–15
Triangle Trigonometry
3 lessons · Right triangles · Law of Sines · Law of Cosines
SOH-CAH-TOAambiguous caselaw selection
+
L13
Right Triangle Trigonometry
SOH-CAH-TOA grounded in the unit circle. Elevation and depression. Bearing problems. Diagram before equation, every time.
L14
Law of Sines and the Ambiguous Case
When Law of Sines applies. The SSA ambiguous case reasoned through — one triangle, two triangles, or none. Finding both solutions when they exist.
L15
Law of Cosines and Area
SAS and SSS cases. Area using (1/2)ab sin C derived from the cross product idea. Choosing which law before solving.
Assignment 5
Solve 8 triangles — 2 right, 2 LoS (one ambiguous), 2 LoC, 2 using area formula. Document method selection before solving.
You solve any triangle and know which tool is right before you start.
06
Unit 6 · Weeks 16–18
Inverse Trig Functions and Applications
2 lessons + capstone · Inverse functions · Applications · Final review
restricted domaininverse compositionreal-world modeling
+
L16
Inverse Trig Functions
Why the domain must be restricted. The range of arcsin, arccos, arctan and how to read them correctly. Compositions like sin(arctan(x)) simplified using triangles.
L17
Real-World Trig Applications
Navigation, physics, and engineering applications requiring trig modeling. Setting up the model before computing. Interpreting the answer in context.
L18 (cap)
Capstone Review and Gap Work
Student selects two weakest units for targeted review. Comprehensive problem set across all six units, reviewed together.
Assignment 6
A timed 75-minute problem set across all 6 units. Every missed problem gets a written correction with reasoning.
You complete trigonometry fluent in every function, identity, equation type, and application.
02Reference
Core Concepts
π rad
Radian Measure
One radian is the angle subtended by an arc equal in length to the radius. Radians are the natural unit for trigonometry — they make calculus and series formulas clean.
π = 180° · 2π = full rotation
sin²+cos²
Pythagorean Identity
sin²θ + cos²θ = 1 follows from the unit circle definition. Divide by cos² to get tan²+1=sec². Divide by sin² to get 1+cot²=csc².
Quotient identities · Reciprocal identities
A sin(Bx+C)+D
Trig Graph Parameters
A is amplitude, 2π/B is period, C/B is phase shift, D is vertical shift. Each parameter changes one feature of the graph independently.
Sin · Cos · Tan graphs
sin(α+β)
Sum Formulas
sin(α+β) = sinαcosβ + cosαsinβ. All double and half angle formulas are derived from these. Knowing the sum formulas means you can derive everything else.
Double angle · Half angle · Product-to-sum
arcsin
Inverse Trig
arcsin(x) returns the angle in [-π/2, π/2] whose sine is x. The domain restriction is necessary to make the inverse a function. The range is fixed, not arbitrary.
Restricted domain · Inverse composition
SSA
The Ambiguous Case
Given two sides and a non-included angle (SSA), there may be 0, 1, or 2 valid triangles. This is why SSA is not a congruence shortcut. The Law of Sines reveals which case applies.
Law of Sines · Law of Cosines · Triangle area
03Capstone
Comprehensive Trig Review

The capstone covers all six units with emphasis on fluency — evaluating, graphing, simplifying, solving, and applying. The goal is to walk into precalculus or physics knowing trig cold.

○
Gap Review
Two weakest units revisited before the problem set.
○
75-Minute Problem Set
Full coverage across all 6 units, timed and independent.
○
Written Corrections
Every missed problem corrected with reasoning, not just the right answer.
○
Precalc Readiness Review
A final session mapping trig skills to what precalculus will demand.
04Investment
Pricing Options
per Hour
$85
1-hr lesson
Flexible · No commitment
Online only
  • Flexible scheduling
  • 1-on-1 focused attention
  • No long-term commitment

Best for trying the curriculum before committing.

Book a session
Popular
per Month
$650
4 lessons · 1.5 hours/lesson
1 lesson/week · book more to move faster
Online + In-Person available
  • Session notes after every lesson
  • Async DM support between sessions
  • Priority scheduling

Booking 2x/week draws down lessons faster — no extra charge.

Get started
Semester
$3,775
26 lessons · 1.5 hours/lesson
~6 months · save vs. monthly
Online + In-Person available
  • Full course arc
  • Session notes after every lesson
  • Capstone review + corrections
  • Next track readiness session

The full arc in one commitment.

Enroll now
In-person sessions available on Monthly and Semester plans at $800/mo and $4,625/semester. NYC only. Per-hour sessions are online only.
05Contact
Let's Talk

Starting trig for the first time or need to fill gaps before precalc? One session is enough to figure out exactly where you are.

Do I need calculus before trig? +
No. Trig comes before calculus — calculus builds on it. You need solid algebra coming in: functions, equations, and some familiarity with graphing.
Is the unit circle memorization-heavy? +
In this course, no. You build the unit circle from two special right triangles in the first session. Once you understand the construction, the values are obvious. Memorization is unnecessary.
Are lessons online or in-person? +
Per-hour sessions are online only. Monthly and Semester plans include the option for in-person sessions in NYC.
What if I need to reschedule? +
Give 24 hours notice and we'll find another time. Monthly and Semester students get priority rebooking.