Law Cosines

Overview

Engineering reference data for Law Cosines 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

Example Calculation

To demonstrate the Law of Cosines, consider finding side c of a triangle where side a = 1 m, side b = 1.3 m, and the included angle C = 60°.

Starting with the formula: $c^2 = a^2 + b^2 - 2ab\cos(C)$

Substitute the known values: $c^2 = (1\ \text{m})^2 + (1.3\ \text{m})^2 - 2(1\ \text{m})(1.3\ \text{m})\cos(60°)$ $c^2 = 1\ \text{m}^2 + 1.69\ \text{m}^2 - (2.6\ \text{m}^2) \times 0.5$ $c^2 = 2.69\ \text{m}^2 - 1.3\ \text{m}^2 = 1.39\ \text{m}^2$ $c = \sqrt{1.39\ \text{m}^2} \approx 1.18\ \text{m}$

The length of side c is approximately 1.18 m.

Supplementary Resources

• A dynamic calculator for the Law of Cosines is available on the source site.
• An Excel template is also provided for performing these calculations offline.

References

• Original Source