The Astronomer's Triangle
Ratios, Functions, Periodicity & The Mathematics of Waves
The Human Problem
Imagine standing on Earth. A mountain is too tall to measure directly. A star is too far away to reach. Yet you can measure an angle. And from that angle, with the right mathematical relationships, you can calculate a distance you cannot touch. Geometry has become a measuring instrument.
TRIANGLE →
CIRCLE →
ANGLES →
RATIOS →
FUNCTIONS →
PERIODICITY →
WAVES →
ROTATION →
PHYSICS
The 53 Trigonometric Cells
From Right Triangles to Periodic Functions
I. Foundations & Ratios
CELL 01-02
The Right Triangle
HypOppAdj
a² + b² = c²
- Opposite, Adjacent, Hypotenuse
- Pythagorean foundation
CELL 04-06
Angle as Rotation
1 rad
- Degrees (360°)
- Radians (2π)
- Arc length relation
CELL 07-13
Trigonometric Ratios
sin =
OH
cos =
AH
tan =
OA
csc =
1sin
- SOH-CAH-TOA
- Reciprocal identities
- Quotient (tan = sin/cos)
CELL 14-15
Special Angles
- 0°, 30°, 45°, 60°, 90°
- Exact values (e.g., √3/2)
II. Unit Circle & Functions
CELL 16
The Four Quadrants
AllSinTanCos
- ASTC rule
- Signs of functions by quadrant
CELL 17-20
Unit Circle & Functions
Domainℝ
Range[-1, 1]
- (cos θ, sin θ)
- f(x) = sin x
- Periodicity (2π)
CELL 21
Trigonometric Graphs
- Sine & Cosine waves
- Tangent asymptotes
- Amplitude & Period
III. Identities & Equations
CELL 22
Pythagorean Identity
sin²x + cos²x = 1
- The foundational identity
- Derived from unit circle
CELL 25-29
Compound Angles
sin(A+B) = sinA cosB + cosA sinB
cos(2A) = cos²A - sin²A
- Sum / Difference formulas
- Double & Half angle
- Product ↔ Sum
CELL 30-33
Trig Equations
sinx = 0
→
x = 0
- Principal vs General solutions
- Using identities to solve
- Identity Proofs (LHS=RHS)
IV. Applications & Oblique Triangles
CELL 34-39
Heights & Distances
- Angle of Elevation / Depression
- Two-point observation
- Inaccessible distances
CELL 40-42
Oblique Triangles
a/sinA = b/sinB
- Law of Sines
- Law of Cosines (Gen. Pythagoras)
- Area = ½ab sinC
V. Frontiers
CELL 43-47
Inverse Trigonometry
sin x = θ
- Reversing the function
- Principal values required
- Domain & Range restrictions
CELL 48-52
Rotation & Waves
- Trig as a wave language
- Amplitude, Frequency, Phase
- Physics & Navigation applications
Exit Portal
We have used angles and circles to measure distances we cannot cross and waves we cannot hold. Now, what happens when we map the number line itself across space? What happens when algebra and geometry finally collide into a single, unified grid?