#G04 · Geometry & Trigonometry · Lv.2View in 3D Mind-Map

Unit Circle Trigonometry & Identities

Hipparchus, Aryabhata & Euler · 140 BCE

cos⁡2θ+sin⁡2θ=1,tan⁡θ=sin⁡θcos⁡θ\cos^2\theta + \sin^2\theta = 1,\quad \tan\theta = \frac{\sin\theta}{\cos\theta}
P(θ)=(cos⁡θ,sin⁡θ),1+tan⁡2θ=sec⁡2θP(\theta) = (\cos\theta, \sin\theta),\quad 1 + \tan^2\theta = \sec^2\theta

Defines cosine and sine as (x, y) coordinates on the unit circle, extending trigonometry to all real angles to model periodic waves, rotations, and oscillations.

🧮 Live Formula Calculator & Plotter
Angle θ°
cos⁡(45∘)=0.707,  sin⁡(45∘)=0.707,  cos⁡2θ+sin⁡2θ=1\cos(45^\circ) = 0.707, \; \sin(45^\circ) = 0.707, \; \cos^2\theta+\sin^2\theta = 1
cos θ (x좌표)
0.7071
sin θ (y좌표)
0.7071
tan θ (기울기)
1.0000
(0.71, 0.71)

Prerequisite Path (6 STEP)

Full Path Analyzer

Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A01Linear Equation & Balance Axiom
STEP 2#A02Polynomial Expansion & Factoring
STEP 3#A03Laws of Exponents & Radicals
STEP 4#G01Pythagorean Theorem
STEP 5#G03Pi, Radian Measure & Circular Sector
STEP 6#G04Unit Circle Trigonometry & Identities

🚀 Next Unlocks (Derived Formulas) (8)

#A09Complex Numbers & Polar Plane#G05Law of Sines & Law of Cosines#G06Trigonometric Addition & Double-Angle Formulas#G08Spherical Coordinates & Sphere Volume/Area#C07Euler's Formula & Euler's Identity#C09Fourier Series & Fourier Transform (FFT)#L033D Cross Product & Surface Normal Vector#L072D/3D Rotation Matrix & Orthogonal Group SO(3)

Geometric & Intuitive Insight

단위원(반지름 1) 위를 도는 점의 그림자를 x축에 비추면 cos θ, y축에 비추면 sin θ가 됩니다. 반지름이 1인 직각삼각형이므로 피타고라스 정리에 의해 언제나 cos²θ + sin²θ = 1² = 1 입니다.

Step-by-Step Derivation

  1. Step 1.1단계: 원점 중심, 반지름 1인 단위원 방정식은 x² + y² = 1 입니다.
  2. Step 2.2단계: 각도 θ에 대한 단위원 위의 점 좌표를 x = cos θ, y = sin θ 로 정의합니다.
  3. Step 3.3단계: 이를 원의 방정식에 대입하면 cos²θ + sin²θ = 1 이며, 양변을 cos²θ로 나누면 1 + tan²θ = sec²θ

AI & Coding Real-World Use

Transformer LLM Sinusoidal/RoPE Positional Encoding and 3D game wave/orbit animation shaders

#LLM RoPE Encoding#Wave Shader#Circular Orbit
const posOnCircle = (r: number, rad: number) => ({ x: r * Math.cos(rad), y: r * Math.sin(rad) });

Geometry & Trigonometry — Related Formulas in This Cluster

Pythagorean Theorem
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Pi, Radian Measure & Circular Sector
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Law of Sines & Law of Cosines
Lv.2 · Inverse Kinematics IK
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Trigonometric Addition & Double-Angle Formulas
Lv.3 · Rotation Matrix Proof
#G06
Conic Sections (Parabola, Ellipse, Hyperbola)
Lv.2 · Kepler Orbit Sim
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