Triangle Solving Formulas | Complete Reference for Sine Law, Cosine Law, and Area

Sine Law (Law of Sines) Formulas

Fundamental relationships between sides and opposite angles

Basic Sine Law

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Fundamental relationship between sides and opposite angles

R is the circumradius of the triangle

Side Calculations

a = 2R\sin A \quad b = 2R\sin B \quad c = 2R\sin C

Calculate sides using angles and circumradius

Useful when circumradius is known

Angle Calculations

\sin A = \frac{a}{2R} \quad \sin B = \frac{b}{2R} \quad \sin C = \frac{c}{2R}

Calculate angle sines using sides and circumradius

Use arcsin to find actual angle values

Ratio Form

a : b : c = \sin A : \sin B : \sin C

Proportional relationship between sides and angle sines

Useful for solving proportion problems

Extended Ratio

\frac{a + b + c}{\sin A + \sin B + \sin C} = 2R

Combined ratio involving all sides and angles

Generalizes the basic sine law relationship

Cosine Law (Law of Cosines) Formulas

Relationships between sides and included angles

Standard Forms

a^2 = b^2 + c^2 - 2bc\cos A

Calculate unknown side using two sides and included angle

Also: b² = a² + c² - 2ac cos B, c² = a² + b² - 2ab cos C

Angle Forms

\cos A = \frac{b^2 + c^2 - a^2}{2bc}

Calculate angle using all three sides

Also: cos B = (a² + c² - b²)/(2ac), cos C = (a² + b² - c²)/(2ab)

Pythagorean Extension

c^2 = a^2 + b^2 \text{ when } C = 90°

Cosine law reduces to Pythagorean theorem for right triangles

cos 90° = 0, eliminating the 2ab cos C term

Triangle Classification

\begin{cases} c^2 < a^2 + b^2 & \text{acute} \\ c^2 = a^2 + b^2 & \text{right} \\ c^2 > a^2 + b^2 & \text{obtuse} \end{cases}

Determine triangle type using side relationships

Applies to the angle opposite the longest side

Triangle Area Formulas

Various methods for calculating triangle area

SAS Area Formula

S = \frac{1}{2}ab\sin C = \frac{1}{2}bc\sin A = \frac{1}{2}ac\sin B

Area using two sides and included angle

Most common formula when angle is known

Heron's Formula

S = \sqrt{s(s-a)(s-b)(s-c)} \text{ where } s = \frac{a+b+c}{2}

Area using only the three sides

s is the semi-perimeter; useful for SSS cases

Base-Height Formula

S = \frac{1}{2} \times \text{base} \times \text{height}

Traditional area formula using base and height

Height is perpendicular distance from vertex to opposite side

Circumradius Area

S = \frac{abc}{4R}

Area using all sides and circumradius

Connects area to circumscribed circle properties

Inradius Area

S = rs = \frac{1}{2}(a+b+c) \cdot r

Area using perimeter and inradius

r is the radius of inscribed circle

Coordinate Area

S = \frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|

Area using vertex coordinates

For vertices A(x₁,y₁), B(x₂,y₂), C(x₃,y₃)

Special Triangle Types

Specialized formulas for specific triangle configurations

Right Triangle

c^2 = a^2 + b^2 \text{ (Pythagorean theorem)}

S = \frac{1}{2}ab \text{ (legs as base and height)}

\sin A = \frac{a}{c}, \cos A = \frac{b}{c}, \tan A = \frac{a}{b}

c is hypotenuse, a and b are legs

Equilateral Triangle

S = \frac{\sqrt{3}}{4}a^2

h = \frac{\sqrt{3}}{2}a \text{ (height)}

R = \frac{a}{\sqrt{3}} \text{ (circumradius)}

r = \frac{a}{2\sqrt{3}} \text{ (inradius)}

All sides equal (a), all angles 60°

Isosceles Triangle

S = \frac{b}{4}\sqrt{4a^2 - b^2}

h = \frac{1}{2}\sqrt{4a^2 - b^2} \text{ (height to base)}

\cos \frac{A}{2} = \frac{a}{b} \text{ (half vertex angle)}

Two equal sides (a), base (b)

Related Geometric Formulas

Additional relationships and properties

Angle Relationships

A + B + C = 180° = \pi \text{ radians}

\sin C = \sin(A + B)

\cos C = -\cos(A + B)

\sin \frac{A+B}{2} = \cos \frac{C}{2}

\cos \frac{A+B}{2} = \sin \frac{C}{2}

Circle Relationships

R = \frac{abc}{4S} \text{ (circumradius)}

r = \frac{S}{s} \text{ (inradius)}

r = (s-a)\tan \frac{A}{2} \text{ (inradius alternative)}

r_a = s\tan \frac{A}{2} \text{ (exradius opposite A)}

Height Formulas

h_a = \frac{2S}{a} = b\sin C = c\sin B

h_b = \frac{2S}{b} = a\sin C = c\sin A

h_c = \frac{2S}{c} = a\sin B = b\sin A

Median Formulas

m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}

m_b = \frac{1}{2}\sqrt{2a^2 + 2c^2 - b^2}

m_c = \frac{1}{2}\sqrt{2a^2 + 2b^2 - c^2}

Triangle Solving Applications

Which formulas to use for different scenarios

Two Angles and One Side (AAS/ASA)

Primary Method: Sine Law

Steps:

1 $Find third angle: C = 180° - A - B$
2 $Use sine law: \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
3 $Calculate unknown sides$

Two Sides and Included Angle (SAS)

Primary Method: Cosine Law → Sine Law

Steps:

1 $Find third side: c² = a² + b² - 2ab cos C$
2 $Find remaining angles using cosine law or sine law$
3 $Calculate area: S = ½ab sin C$

All Three Sides (SSS)

Primary Method: Cosine Law

Steps:

1 $Find angles: cos A = (b² + c² - a²)/(2bc)$
2 $Repeat for other angles$
3 $Calculate area using Heron's formula$

Two Sides and Non-Included Angle (SSA)

Primary Method: Sine Law (Ambiguous Case)

Steps:

1 $Use sine law: sin B = (b sin A)/a$
2 $Check for 0, 1, or 2 solutions$
3 $Find remaining elements for each valid solution$

Quick Reference Guide

Essential formulas at a glance

Given Information	Primary Formula	Additional Notes
Two angles, one side (AAS/ASA)	$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$	Unique solution
Two sides, included angle (SAS)	$c^2 = a^2 + b^2 - 2ab\cos C$	Use cosine law first
All three sides (SSS)	$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$	Use cosine law for angles
Two sides, opposite angle (SSA)	$\sin B = \frac{b\sin A}{a}$	Ambiguous case: 0, 1, or 2 solutions
Area (SAS known)	$S = \frac{1}{2}ab\sin C$	Most direct area method
Area (SSS known)	$S = \sqrt{s(s-a)(s-b)(s-c)}$	Heron's formula