Trigonometry is the part of mathematics that connects angles with lengths. It often begins with right triangles, where sine, cosine, and tangent describe the relationship between an angle and the sides around it. The same ideas later extend to circles, graphs, waves, rotation, and other repeating patterns.

At the center of introductory trigonometry are three ratios:

sin θ = opposite / hypotenuse

cos θ = adjacent / hypotenuse

tan θ = opposite / adjacent

Once those relationships make sense, the rest of the subject becomes much easier to connect.

Start with a Right Triangle

A right triangle contains one 90° angle. Choose either of the other two angles and call it θ.

The longest side is the hypotenuse. It always sits opposite the 90° angle. The other two names depend on the angle you chose:

• The opposite side is directly across from θ.
• The adjacent side is next to θ, but is not the hypotenuse.

If you switch to the other acute angle, the opposite and adjacent sides switch too. The hypotenuse does not change.

Right triangle and SOHCAHTOA: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, and tan θ = opposite / adjacent.

Sine, Cosine, and Tangent

The three basic trigonometric functions compare pairs of sides.

Function	Ratio
Sine	sin θ = opposite / hypotenuse
Cosine	cos θ = adjacent / hypotenuse
Tangent	tan θ = opposite / adjacent

SOHCAHTOA is a compact way to remember them:

SOH — Sine = Opposite / Hypotenuse
CAH — Cosine = Adjacent / Hypotenuse
TOA — Tangent = Opposite / Adjacent

Choosing a function is less about memorizing rules and more about noticing which sides appear in the problem. If the opposite side and hypotenuse are involved, use sine. If the adjacent side and hypotenuse are involved, use cosine. If the opposite and adjacent sides are involved, use tangent.

A Simple Trigonometry Example

Suppose a right triangle has a 30° angle and a hypotenuse of 10. We want to find the side opposite the 30° angle.

The opposite side and hypotenuse point us to sine:

sin 30° = opposite / 10

Because sin 30° = 1/2:

1/2 = opposite / 10

Multiply both sides by 10:

opposite = 5

The missing side is 5 units long.

Most right-triangle problems follow the same pattern: identify what is known, decide what must be found, choose the ratio that connects them, and solve.

Why the Ratios Stay the Same

Imagine two right triangles that both contain a 30° angle. One might be small enough to draw on a page; the other might represent the height of a building. Their side lengths are different, but their angles match, so the triangles are similar.

Similar triangles have the same proportions. That is why sin 30° is always 1/2, no matter how large the triangle is.

A trigonometric function does not describe the size of one particular triangle. It describes a relationship created by an angle.

The Six Trigonometric Functions

There are six standard trigonometric functions. Sine, cosine, and tangent are used most often at the introductory level; cosecant, secant, and cotangent are their reciprocals.

Function	Right-triangle definition
sin θ	opposite / hypotenuse
cos θ	adjacent / hypotenuse
tan θ	opposite / adjacent
csc θ	hypotenuse / opposite
sec θ	hypotenuse / adjacent
cot θ	adjacent / opposite

The reciprocal relationships are:

csc θ = 1 / sin θ
sec θ = 1 / cos θ
cot θ = 1 / tan θ

Tangent can also be written as:

tan θ = sin θ / cos θ

For a first pass through trigonometry, sine, cosine, and tangent do most of the work. The reciprocal functions become more useful when you study identities and equations.

Special Right Triangles and Common Angles

A few angles appear so often that their exact values are worth recognizing. Those values come from two special right triangles rather than from an arbitrary table.

A 45°–45°–90° triangle has side lengths in the ratio:

1 : 1 : √2

A 30°–60°–90° triangle has side lengths in the ratio:

1 : √3 : 2

Special right triangles explain the exact trigonometric values of 30°, 45°, and 60°.

From the 30°–60°–90° triangle, for example:

sin 30° = 1/2
cos 30° = √3/2
tan 30° = √3/3

From the 45°–45°–90° triangle:

sin 45° = cos 45° = √2/2
tan 45° = 1

Here are the five common first-quadrant values together:

Angle	Radians	sin θ	cos θ	tan θ
0°	0	0	1	0
30°	π/6	1/2	√3/2	√3/3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	Undefined

From Triangles to the Unit Circle

Right triangles give a clear introduction, but they do not by themselves explain angles such as 150°, 270°, −45°, or an angle that makes several complete turns. The unit circle extends sine and cosine to any angle.

A unit circle is centered at (0, 0) and has radius 1. Start on the positive x-axis and rotate through an angle θ. The point where the terminal side meets the circle has coordinates:

(cos θ, sin θ)

That means cosine is the point’s horizontal coordinate and sine is its vertical coordinate.

On the unit circle, cos θ is the x-coordinate and sin θ is the y-coordinate.

At 0°, the point is (1, 0), so cos 0° = 1 and sin 0° = 0.

At 90°, the point is (0, 1), so cos 90° = 0 and sin 90° = 1.

At 180°, the point is (−1, 0), so cos 180° = −1 and sin 180° = 0.

The coordinate signs explain the signs of the three main functions:

Quadrant	sin	cos	tan
I	+	+	+
II	+	−	−
III	−	−	+
IV	−	+	−

Tangent follows from tan θ = sin θ / cos θ. It is positive when sine and cosine have the same sign, and negative when their signs differ.

Degrees and Radians

Angles are usually measured in degrees or radians. A complete turn can be written either way:

360° = 2π radians

So:

180° = π radians
90° = π/2 radians

To convert degrees to radians:

radians = degrees × π / 180

To convert radians to degrees:

degrees = radians × 180 / π

Degrees	Radians
0°	0
30°	π/6
45°	π/4
60°	π/3
90°	π/2
180°	π
270°	3π/2
360°	2π

Degrees are often easier to picture when you first learn about angles. Radians become especially important in calculus, physics, engineering, and any work involving periodic functions.

The Pythagorean Identity

The most important trigonometric identity is:

sin² θ + cos² θ = 1

It comes directly from the unit circle. Every point (x, y) on a circle of radius 1 satisfies:

x² + y² = 1

Since x = cos θ and y = sin θ, substituting those coordinates gives the Pythagorean identity.

Two related identities are:

1 + tan² θ = sec² θ
1 + cot² θ = csc² θ

These relationships are useful when you know one trigonometric value and need another, or when an expression needs to be simplified.

Finding an Angle with Inverse Trigonometry

Trigonometric functions can also work in reverse. Suppose the opposite and adjacent sides of a right triangle are both 5. Then:

tan θ = 5 / 5 = 1

To recover the angle, use inverse tangent:

θ = tan⁻¹(1) = 45°

The three inverse functions used most often are:

• sin⁻¹ or arcsin finds an angle from its sine.
• cos⁻¹ or arccos finds an angle from its cosine.
• tan⁻¹ or arctan finds an angle from its tangent.

One notation detail matters: sin⁻¹(x) means inverse sine here. It does not mean 1 / sin(x). The reciprocal of sine is csc(x).

Trigonometry Beyond Right Triangles

Not every triangle contains a right angle. Two formulas extend trigonometry to general triangles.

Law of Sines

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

a / sin A = b / sin B = c / sin C

The Law of Sines is useful when you know two angles and one side, or two sides and an angle opposite one of them.

Law of Cosines

c² = a² + b² − 2ab cos C

The Law of Cosines is useful when you know two sides and the included angle, or all three sides.

The Pythagorean theorem is a special case. If C = 90°, then cos 90° = 0, leaving:

c² = a² + b²

Trigonometric Graphs

Sine, cosine, and tangent can also be treated as functions. Their graphs show how each value changes continuously as the angle changes.

Sine and cosine repeat every 2π. Tangent repeats every π and is undefined at its vertical asymptotes.

Sine and cosine both produce smooth repeating waves. Their period is:

360° = 2π radians

Their values always remain between −1 and 1:

−1 ≤ sin θ ≤ 1
−1 ≤ cos θ ≤ 1

Tangent behaves differently. Because tan θ = sin θ / cos θ, it is undefined wherever cos θ = 0. Its period is shorter:

180° = π radians

Why Sine and Cosine Form Waves

Imagine a point moving around a circle at a steady speed. Its vertical position rises, falls, and repeats. If you plot that vertical position against time, you get a sine wave. Its horizontal position follows a cosine wave.

That link between circular motion and waves is one of the central ideas in trigonometry. It is also why the same functions appear in sound, light, alternating current, vibrations, rotating machinery, and signal processing.

A Real-World Example

Suppose you stand 50 meters from a building and measure an angle of elevation of 40° to the top. The ground, the building, and your line of sight form a right triangle.

The height is opposite the 40° angle, and the 50-meter distance is adjacent, so tangent is the useful ratio:

tan 40° = height / 50

Rearrange the equation:

height = 50 × tan 40°

The result is about 42 meters. Trigonometry lets you estimate the height without measuring the wall directly.

Where Trigonometry Is Used

Trigonometry is useful wherever angles, distance, rotation, or repeating motion matter.

Surveying uses angles to calculate distances, slopes, elevations, and boundaries.

Navigation uses bearings, positions, coordinates, and distances.

Physics uses trigonometry with vectors, forces, circular motion, oscillations, and waves.

Engineering applies it to structures, machines, electronics, and signals.

Astronomy uses angles to describe the positions and movement of celestial objects.

Computer graphics relies on sine and cosine for rotation, movement, cameras, animation, and 3D positioning.

Sound and signal processing use combinations of sine and cosine waves to represent and analyze complex signals.

Trigonometry at a Glance

Concept	Formula
Sine	sin θ = opposite / hypotenuse
Cosine	cos θ = adjacent / hypotenuse
Tangent	tan θ = opposite / adjacent
Cosecant	csc θ = 1 / sin θ
Secant	sec θ = 1 / cos θ
Cotangent	cot θ = 1 / tan θ
Pythagorean identity	sin² θ + cos² θ = 1
Tangent identity	tan θ = sin θ / cos θ
Unit-circle point	(cos θ, sin θ)
Full rotation	360° = 2π radians
Law of Sines	a / sin A = b / sin B = c / sin C
Law of Cosines	c² = a² + b² − 2ab cos C

If you are new to trigonometry, start with right triangles and learn how to identify the opposite, adjacent, and hypotenuse sides. Then learn sine, cosine, tangent, and SOHCAHTOA. From there, the unit circle connects those same ideas to any angle, and the graphs show how they behave as functions.

Trigonometry table

Trigonometric tables are convenient tools for querying trigonometric values. This chapter distinguishes between basic standard tables (covering common angles such as 0°, 30°, 45°, etc.) and scientific tables (covering the full 0°-360° range, high-precision values, and special angles). They not only meet the needs of beginners for quick queries of basic angle values but also provide high-precision numerical support for fields such as physics, engineering, and astronomy. Meanwhile, the patterns in the table values help learners deepen their understanding of properties like periodicity and monotonicity of trigonometric functions.

Basic standard tables of Trigonometry

The trigonometric table is simply a collection of the values of trigonometric ratios for various standard angles including 0°, 30°, 45°, 60°, 90°, sometimes with other angles like 180°, 270°, and 360° included, in a tabular format. Because of patterns existing within trigonometric ratios and even between angles, it is easy to both predict the values of the trigonometry table and use the table as a reference to calculate trigonometric values for various other angles. The trigonometric functions are namely the sine function, cosine function, tan function, cot function, sec function, and cosec function.

Scientific tables of Trigonometry

---