Geometric laws

Statement

Triangle inequality: If $a$, $b$, and $c$ are the lengths of the sides of a triangle then the triangle inequality states that

$$c \leq a + b,$$

with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about vectors and vector lengths (norms):

$$\|\mathbf u + \mathbf v\| \leq \|\mathbf u\| + \|\mathbf v\| ,$$

where the length of the third side has been replaced by the length of the vector sum $\mathbf{u} + \mathbf{v}$. When $\mathbf{u}$ and $\mathbf{v}$ are real numbers, they can be viewed as vectors in $\mathbb{R}^1$, and the triangle inequality expresses a relationship between absolute values.

Pythagorean theorem: It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation:

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

Geometrically, trigonometric identities are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities involving both angles and side lengths of a triangle. Only the former are covered in this article.

These identities are useful whenever expressions involving trigonometric functions need to be simplified. Another important application is the integration of non-trigonometric functions: a common technique which involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

One of the most prominent examples of trigonometric identities involves the equation $\sin^2 \theta + \cos^2 \theta = 1,$ which is true for all real values of $\theta$. On the other hand, the equation

$$\cos\theta = 1$$

is only true for certain values of $\theta$, not all. For example, this equation is true when $\theta = 0,$ but false when $\theta = 2$.

Another group of trigonometric identities concerns the so-called addition/subtraction formulas (e.g. the double-angle identity $\sin(2\theta) = 2\sin\theta \cos\theta$, the addition formula for $\tan(x + y)$), which can be used to break down expressions of larger angles into those with smaller constituents.