#G05 · Geometry & Trigonometry · Lv.2View in 3D Mind-Map

Law of Sines & Law of Cosines

Al-Kashi & François Viète · 1427 CE

c^2 = a^2 + b^2 - 2ab\cos C,\quad \frac{a}{\sin A} = \frac{b}{\sin B} = 2R

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

Generalizes the Pythagorean theorem to any triangle via the -2ab cos C correction term, serving as the geometric parent of the vector dot product.

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Side a

Side b

Angle C°

c = \sqrt{7^2 + 8^2 - 2(7)(8)\cos 60^\circ} = 7.550

Opposite Side c

7.5498

Triangle Area

24.249

Circumdiameter 2R

8.718

Prerequisite Path (7 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#G03Pi, Radian Measure & Circular Sector

l = r\theta

STEP 5#G01Pythagorean Theorem

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

STEP 6#G04Unit Circle Trigonometry & Identities

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

STEP 7#G05Law of Sines & Law of Cosines

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

🚀 Next Unlocks (Derived Formulas) (2)

#G06Trigonometric Addition & Double-Angle Formulas #L02Dot Product & Cosine Similarity

Geometric & Intuitive Insight

각 C가 90°일 때 cos 90° = 0 이므로 c² = a² + b² (피타고라스 정리)이 되고, 예각이면 빗변이 줄어들어 -2ab cos C 만큼 작아지며, 둔각이면 cos C < 0 이라 더 길어집니다.

Step-by-Step Derivation

Step 1.1단계: 꼭짓점 C를 원점 (0,0), 점 B를 (a,0)에 두면 점 A의 좌표는 (b cos C, b sin C) 입니다.
Step 2.2단계: 두 점 A, B 사이의 거리 제곱 c² = (b cos C - a)² + (b sin C - 0)² 을 전개합니다.
Step 3.3단계: b²(cos²C + sin²C) - 2ab cos C + a² = a² + b² - 2ab cos C 가 유도됩니다.

AI & Coding Real-World Use

Robotics & 3D character Inverse Kinematics (IK) joint angle solver and GPS triangulation

#Inverse Kinematics IK#GPS Triangulation#Vector Dot Product

const cosC = (a*a + b*b - c*c) / (2 * a * b);

Geometry & Trigonometry — Related Formulas in This Cluster

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