Trigonometry

In geometry, trigonometry is the study of the trigonometric functions, functions that seek to relate the lengths and angles of triangles. Trigonometry is integral to geometry, as many famous results were proven using trigonometry.

In contest math, trigonometry's use is not just limited to geometry; problems involving equations with trigonometric functions are very common. These are often solved via clever usage of the trigonometric functions' many identities, which drastically simplify expressions.

Outside of competition math, trigonometry is the backbone of much of analysis, especially Fourier analysis.

Definitions

The trigonometric functions have several definitions. The definition usually taught first is the right triangle definition, for its ease of access. An intermediate to olympiad geometry course usually uses the unit circle definition of trigonometry. In complex analysis, the Taylor series definition of trigonometry is preferred to extend trigonometry to a complex domain, although this is beyond the scope of competition math.

Right triangle definition

The right triangle definition of trigonometry involves the ratios between edges of a right triangle, with respect to a given angle. The definitions below will be referring to angle $A$ , with side lengths specified in the diagram. Because angle $A$ must be less than $90^{\circ}$ for the triangle to stay right, these definitions only work for acute angles.

Sine: The sine of angle $A$ , denoted $\sin (A)$ , is defined as the ratio of the side opposite $A$ to

the hypotenuse. $\sin (A) = \frac{\textrm{opposite}}{\textrm{hypotenuse}} = \frac{a}{c}.$
Cosine: The sine of angle $A$ , denoted $\sin (A)$ , is defined as the ratio of the side adjacent $A$ to the hypotenuse. $\cos (A) = \frac{\textrm{adjacent}}{\textrm{hypotenuse}} = \frac{b}{c}.$
Tangent: The tangent of angle $A$ , denoted $\tan (A)$ , is defined as the ratio of the side opposite $A$ to the side adjacent to $A$ . $\tan (A) = \frac{\textrm{opposite}}{\textrm{adjacent}} = \frac{a}{b}.$

A common mneumonic to remember this is SOH-CAH-TOA, where Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, and Tangent = Opposite / Adjacent

More uncommon are the reciprocals of the trigonometric functions, listed below.

Cosecant: The cosecant of angle $A$ , denoted $\csc (A)$ , is defined as the reciprocal of the sine of $A$ . $\csc (A) = \frac{1}{\sin (x)} = \frac{\textrm{hypotenuse}}{\textrm{opposite}} = \frac{c}{a}.$
Secant: The secant of angle $A$ , denoted $\sec (A)$ , is defined as the reciprocal of the cosine of $A$ . $\sec (A) = \frac{1}{\cos (x)} = \frac{\textrm{hypotenuse}}{\textrm{adjacent}} = \frac{c}{b}.$
Cotangent: The cotangent of angle $A$ , denoted $\cot (A)$ , is defined as the reciprocal of the tangent of $A$ . $\cot (A) = \frac{1}{\tan (x)} = \frac{\textrm{adjacent}}{\textrm{opposite}} = \frac{b}{a}.$

This definition is most commonly taught in introductory geometry classes for its simplicity. However, it is limiting. It only works if $\triangle ABC$ is right, which means that the trigonometric functions are only defined when angle $A$ is acute.

Unit circle definition

Consider the unit circle, the circle with radius one centered at the origin. Starting at $(1, 0)$ , walk a distance $\theta$ counterclockwise around the unit circle, as shown in the diagram. The coordinates of this point are defined to be $(\cos (\theta), \sin (\theta) )$ .

As for the other trigonometric functions, $\tan (\theta)$ is defined to be the ratio of $\sin (\theta)$ to $\cos (\theta)$ , and cosecant, secant, and cotangent are defined to be the reciprocals of sine, cosine, and tangent, respectively.

The benefit of this definition is that it matches the right triangle definition for acute angles, but extends their domain from acute angles to all real-valued angles. As such, this definition is usually preferred in intermediate to olympiad geometry settings.

Taylor series definition

The Taylor series for sine and cosine are used as the definition of sine and cosine in analysis, particularly complex analysis. Defining the trigonometric functions this way gives a concrete way to extend the definition of trigonometry from the real numbers to the full complex plane. The taylor series for sine and cosine are shown below: $\sin (x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \cdots$ $\cos (x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \cdots$ These definitions are not used in high school math competitions; however, they do appear in the Putnam and other university competitions.

Applications to geometry

Trigonometry sees significant use in intermediate geometry. Aside from returning ratios of right triangles, trigonometry can apply to all triangles, via the law of sines and the law of cosines.

Law of sines

The law of sines states that in any $\triangle ABC$ , $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R,$ where $a$ is the side opposite to $A$ , $b$ opposite to $B$ , $c$ opposite to $C$ , and and $R$ is the circumradius of $\triangle ABC$ . The law of sines is particularly handy in problems involving the circumradius, seeing extremely wide usage in intermediate geometry.

Law of cosines

The law of cosines states that in any $\triangle ABC$ , $c^2 = a^2 + b^2 - 2ab\cos (C),$ where $a$ is the side opposite to $A$ , $b$ opposite to $B$ , and $c$ opposite to $C$ . It is a generalization of the Pythagorean theorem and is used to prove several famous results, such as Heron's formula and Stewart's theorem. However, it sees limited applicability compared to the law of sines, as usage of the law of cosines can get very algebra-heavy.

Trigonometric identities

Trigonometric identities are expressions involving trigonometric functions that are true for all inputs. The trigonometric functions are known for their identities. The most commonly used identities in contest math are:

Pythagorean identities
Angle addition identities
Double angle identities
Half angle identities
Sum-to-product identities
Product-to-sum identities

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