Trigonometry: Sine, Cosine, Tangent and the Basic Formulas

What is trigonometry

Trigonometry is the branch of math that studies the relationship between a triangle’s angles and the length of its sides. The name comes from Greek, “trigonon” (triangle) and “metron” (measure): literally, measuring triangles. According to Encyclopaedia Britannica, the field emerged in the Hellenistic world in the 3rd century BC, when early geometers applied the tools of geometry to problems in astronomy. In algebra you work with letters instead of numbers; here a Greek letter like θ (theta) stands for an angle, and the formulas linking it to the sides work the same way for any right triangle, small or enormous.

Until roughly the 16th century, Britannica notes, trigonometry had one central problem: given some parts of a triangle, work out all the others. It’s the same practical goal behind anyone today who surveys a plot of land, plots a course, or observes a star.

Right triangle trigonometry: sine, cosine and tangent (SOHCAHTOA)

In a right triangle, every acute angle θ has three reference sides: the hypotenuse (the longest side, opposite the right angle), the leg opposite θ, and the leg adjacent to θ. These three sides give the three basic ratios, confirmed in identical terms by Wolfram MathWorld and the OpenStax university textbook:

Ratio	Definition	Mnemonic
Sine (sin θ)	opposite leg ÷ hypotenuse	SOH
Cosine (cos θ)	adjacent leg ÷ hypotenuse	CAH
Tangent (tan θ)	opposite leg ÷ adjacent leg	TOA

The initials of the three lines spell SOHCAHTOA, the mnemonic generations of students have used to learn the three definitions. No other numbers are needed to get started: three ratios and one angle are enough.

From the Pythagorean theorem to the basic identities

The three definitions aren’t independent of each other. Draw a right triangle with a hypotenuse of exactly 1, and the Pythagorean theorem says the sum of the squares of the two legs equals the square of the hypotenuse. Since in that triangle the opposite leg measures sin θ and the adjacent leg measures cos θ, the same equality reads:

sin²θ + cos²θ = 1

That’s the Pythagorean identity, the fundamental relationship in trigonometry. Two more identities follow from it: tan²θ + 1 = sec²θ and 1 + cot²θ = csc²θ, where sec, csc and cot are the reciprocals of cosine, sine and tangent. Anyone who has already studied Euclidean geometry will recognize the same Pythagorean theorem here, just applied differently.

The special values: 30°, 45°, 60°

A handful of angles come up so often that they have exact values, worked out without a calculator. OpenStax derives them from two special triangles: the isosceles 45-45-90, and the 30-60-90, obtained by cutting an equilateral triangle in half.

Angle	sin	cos	tan
30°	1/2	√3/2	√3/3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3

These are fixed geometric properties, not figures that change over time: you can always redraw the two triangles and check them row by row.

Practical example: someone walks away from the base of a tree to a point 30 feet away and, from there, measures a 57° angle up to the top with a clinometer. The tree’s height is then h = 30 × tan(57°), about 46 feet, a little over 14 meters: the tangent turns a distance you can measure from the ground into a height nobody has to climb to find out. The example, with these same numbers, comes from the OpenStax textbook “Precalculus 2e”.

Trigonometry beyond the triangle: stellar parallax

The same principle works on a cosmic scale. To measure the distance to a nearby star, astronomers observe it twice, six months apart, from opposite sides of Earth’s orbit: the baseline of this enormous triangle is the diameter of that orbit, about 2 astronomical units. Against the backdrop of much more distant stars, the nearby star appears to shift by a tiny angle, stellar parallax, and that angle, through the tangent, gives the distance: d = 1 AU / tan(p), according to Encyclopaedia Britannica.

When the parallax angle measures exactly one arcsecond, the corresponding distance is 206,265 astronomical units. That distance has its own name, the parsec, and it’s still the unit astronomers use for stars within our galaxy.

Navigation, surveying, and a common mix-up: GPS

For centuries, trigonometry guided anyone who had to find their way without fixed landmarks: triangulation, measuring angles from two known points to work out the position of a third, underlies mapmaking and both sea and air navigation, as well as land surveying. It’s the same principle at work when solving equations for a triangle where only some parts are known.

GPS, though, doesn’t work that way. According to GPS.gov, the official U.S. government site for the system, each receiver measures how long a radio signal takes to arrive from at least three satellites, a signal traveling at the speed of light, 299,792,458 meters per second, and turns that time into a distance (Distance = Rate × Time). With distances from three satellites, it pins down a point in space: that’s trilateration, not triangulation, because the calculation starts from distances rather than angles. The same distinction matters for orbits and measurements in the solar system.