#G06 · Geometry & Trigonometry · Lv.3View in 3D Mind-Map

Trigonometric Addition & Double-Angle Formulas

Claudius Ptolemy · 150 CE

\cos(\alpha\pm\beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta

\sin(\alpha\pm\beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta

Expands sine and cosine of angle sums (α ± β) into products of single-angle functions—essential for proving trig derivatives, 2D rotation matrices, and RF modulation.

🧮 Live Formula Calculator & Plotter

Angle α°

Angle β°

\sin(45^\circ+30^\circ) = 0.9659, \quad \cos(45^\circ+30^\circ) = 0.2588

sin(α+β) = sin(75°)

0.9659

cos(α+β) = cos(75°)

0.2588

배각 sin(2α)

1.0000

Prerequisite Path (9 STEP)

Full Path Analyzer

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

STEP 1#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 2#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 3#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 4#G01Pythagorean Theorem

a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}

STEP 5#G03Pi, Radian Measure & Circular Sector

l = r\theta

STEP 6#G022D/3D Euclidean Distance Formula

d(P

STEP 7#G04Unit Circle Trigonometry & Identities

\cos^2\theta + \sin^2\theta = 1

STEP 8#G05Law of Sines & Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos C

STEP 9#G06Trigonometric Addition & Double-Angle Formulas

\cos(\alpha\pm\beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta

🚀 Next Unlocks (Derived Formulas) (2)

#C09Fourier Series & Fourier Transform (FFT)#L072D/3D Rotation Matrix & Orthogonal Group SO(3)

Geometric & Intuitive Insight

단위원 위의 두 점 P(cos α, sin α)와 Q(cos β, sin β) 사이의 거리를 유클리드 거리 공식으로 구한 값과, 끼인각 (α-β)에 대해 제2코사인법칙으로 구한 값을 같다고 놓으면 즉시 증명됩니다.

Step-by-Step Derivation

Step 1.1단계: 단위원 위 두 점 A(cos α, sin α), B(cos β, sin β) 사이의 거리 제곱은 (cos α - cos β)² + (sin α - sin β)² = 2 - 2(cos α cos β + sin α sin β)
Step 2.2단계: 한편 원점과 두 점 A, B가 이루는 삼각형에서 제2코사인법칙을 쓰면 AB² = 1² + 1² - 2·1·1·cos(α - β) = 2 - 2cos(α - β)
Step 3.3단계: 두 식을 비교하면 cos(α - β) = cos α cos β + sin α sin β 가 성립합니다.

AI & Coding Real-World Use

Deriving 2D/3D rotation matrix composition R(α)R(β)=R(α+β) and RF QAM signal demodulation

#Rotation Matrix Proof#5G QAM Modulation#Audio DSP

const sinAdd = (a: number, b: number) => Math.sin(a)*Math.cos(b) + Math.cos(a)*Math.sin(b);

Geometry & Trigonometry — Related Formulas in This Cluster

Pythagorean Theorem

Lv.1 · Game Physics Distance

#G01

2D/3D Euclidean Distance Formula

Lv.2 · K-Means & KNN ML

#G02

Pi, Radian Measure & Circular Sector

Lv.1 · Three.js Radians

#G03

Unit Circle Trigonometry & Identities

Lv.2 · LLM RoPE Encoding

#G04

Law of Sines & Law of Cosines

Lv.2 · Inverse Kinematics IK

#G05

Conic Sections (Parabola, Ellipse, Hyperbola)

Lv.2 · Kepler Orbit Sim

#G07