Trigonometry

Trigonometry is a branch of mathematics concerned with relationships between angles and lengths, especially the sides of triangles, and with the functions that express those relationships. It connects geometry with algebra by turning geometric measurements into equations. Although its elementary definitions use right triangles, its modern scope includes circular functions defined for arbitrary real arguments and methods for describing periodic behavior. Its development was closely connected with astronomy, where angular observations were used to calculate otherwise inaccessible distances and positions. (openstax.org)

Historical development

Early trigonometry employed chords: straight segments joining two points on a circle. In ancient Greece, Hipparchus produced a chord table around 140 BCE, although the table itself has not survived. In the second century CE, Ptolemy developed methods for calculating chords using inscribed polygons, half-angle relationships, and interpolation. These calculations served astronomical models rather than a separate subject resembling a modern trigonometry course. (mathshistory.st-andrews.ac.uk)

Indian mathematicians shifted attention toward half-chords, the geometric predecessors of the sine function. Around 500 CE, Aryabhata provided tables based on these quantities. Scholars writing in Arabic subsequently developed trigonometric tables and relationships further, and this mathematical tradition later influenced European work. Historically, calculations involving spherical triangles were important alongside those involving plane triangles, particularly because astronomical positions were represented on a sphere. (mathshistory.st-andrews.ac.uk)

Angles and right-triangle ratios

Angles are commonly measured in degrees or radians. A complete turn is $360^\circ$, or $2\pi$ radians. Radian measure defines an angle as the ratio of the intercepted arc length to the circle’s radius, making it especially suitable for mathematical analysis. Positive angles conventionally represent counterclockwise rotation; negative angles represent clockwise rotation. (openstax.org)

For an acute angle $\theta$ in a right triangle, the principal trigonometric functions are defined by

Here the hypotenuse is the side opposite the right angle, while “opposite” and “adjacent” refer to the chosen acute angle. Similar triangles have identical corresponding side ratios, so these values depend on the angle rather than the triangle’s size. The remaining standard functions are reciprocals: cosecant is $1/\sin\theta$, secant is $1/\cos\theta$, and cotangent is $1/\tan\theta$, wherever the denominators are nonzero. (openstax.org)

The unit circle and periodicity

The unit circle, centered at the origin with radius one, extends the definitions beyond acute angles. If an angle $\theta$, measured from the positive horizontal axis, identifies a point $(x,y)$ on this circle, then

These definitions apply to every real number $\theta$, including negative values and angles exceeding one revolution. Tangent is $\sin\theta/\cos\theta$ and is undefined at $\theta=\pi/2+k\pi$, where $k$ is an integer. (openstax.org)

Sine and cosine are periodic functions with fundamental period $2\pi$; tangent has fundamental period $\pi$. Both sine and cosine take values in $[-1,1]$. Their graphs form smooth repeating curves, whereas tangent has vertical asymptotes. The Pythagorean theorem, applied to the unit circle, gives the fundamental identity

Thus the circle provides a unified explanation of function values, signs, and repetition. (openstax.org)

Identities and equations

A trigonometric identity is an equality valid for every argument for which its expressions are defined. Important examples are the addition formulas:

Setting $\alpha=\beta$ yields double-angle formulas, including $\sin(2\alpha)=2\sin\alpha\cos\alpha$. Such identities allow expressions to be transformed, exact values to be calculated, and equations to be solved. Unlike an identity, a trigonometric equation generally holds only for particular arguments; periodicity often produces infinitely many solutions. (openstax.org)

Solving general triangles

For a plane triangle with sides $a,b,c$ opposite angles $A,B,C$, the law of sines states

It is useful when two angles and one side are known. When two sides and an angle opposite one of them are specified, the data can describe no triangle, one triangle, or two triangles—the ambiguous case. (openstax.org)

The law of cosines states

It determines the third side from two sides and their included angle, or angles from three sides. When $C=90^\circ$, it reduces to the Pythagorean theorem. A related area formula is $K=\tfrac12 ab\sin C$. Together these relations make trigonometry a practical method for reconstructing triangles from partial measurements. (openstax.org)

Calculus and measurement

In calculus, using radian arguments gives particularly simple derivatives:

These relationships make trigonometric functions central to models of oscillation in physics. For example, $y(t)=A\cos(\omega t+\phi)$ represents simple harmonic motion, with amplitude $A$, angular frequency $\omega$, and phase $\phi$; differentiation supplies its velocity and acceleration. (openstax.org)

Trigonometry also converts angular observations into lengths. On level ground, an object’s height above an observer’s eye level is $d\tan\theta$, where $d$ is the horizontal distance and $\theta$ the elevation angle. The observer’s eye height must be added to obtain the object’s total height. This illustrates how triangle relationships support indirect measurement without requiring access to an object’s top. (openstax.org)