Law Tangents

Overview

Engineering reference data for Law Tangents in mathematics.

Key Formulas

Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Roots of ax² + bx + c = 0.

Pythagorean Theorem

$c^2 = a^2 + b^2$

Right triangle relationship.

Circle Area

$A = \pi r^2$

Area of a circle.

Logarithm

$\log_b(x) = \frac{\ln(x)}{\ln(b)}$

Change of base formula.

Variables

Derivation and Properties

The Law of Tangents is a trigonometric identity relating the sides and angles of a triangle. It can be derived from the Law of Sines.

For a triangle with sides $a$, $b$, $c$ and opposite angles $A$, $B$, $C$:

$\frac{a + b}{a - b} = \frac{\tan\left(\frac{A + B}{2}\right)}{\tan\left(\frac{A - B}{2}\right)}$

This identity is particularly useful for solving triangles when you know two sides and the included angle, or two angles and a side.

Relationship to Other Triangle Laws

The Law of Tangents exists alongside other fundamental triangle relationships:

• Pythagorean Theorem (for right triangles): $a^2 + b^2 = c^2$
• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Angle Calculation from Side Lengths

If all three side lengths ($a$, $b$, $c$) are known, the angles can be calculated using the Law of Cosines: $A = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right)$ $B = \arccos\left(\frac{a^2 + c^2 - b^2}{2ac}\right)$ $C = \arccos\left(\frac{a^2 + b^2 - c^2}{2ab}\right)$

References

• Original Source