What Is Trigonometry, Really?
Trigonometry is the mathematics of triangles: specifically, the fixed relationships between a triangle's angles and its side lengths. If you know one angle and one side of a right triangle, you can work out everything else without measuring. That single idea underpins surveying, navigation, construction, physics, and every 3D video game ever made.
The whole subject rests on three functions: sine, cosine, and tangent. Each one takes an angle and returns the ratio of two sides.
The Three Core Functions (SOH-CAH-TOA)
Take a right triangle. Pick one of the non-right angles and call it θ. The side opposite θ is the "opposite", the side next to it (that isn't the hypotenuse) is the "adjacent", and the longest side is the "hypotenuse". Then:
- sin(θ) = Opposite ÷ Hypotenuse (SOH)
- cos(θ) = Adjacent ÷ Hypotenuse (CAH)
- tan(θ) = Opposite ÷ Adjacent (TOA)
The mnemonic SOH-CAH-TOA (pronounced "so-ka-toa") is how generations of students have remembered this. Note that tan(θ) is also sin(θ) ÷ cos(θ): the three functions aren't independent.
Key Reference Values
These five angles appear constantly. Worth memorising, or at least recognising:
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 0.5 | √3/2 ≈ 0.866 | 1/√3 ≈ 0.577 |
| 45° | √2/2 ≈ 0.707 | √2/2 ≈ 0.707 | 1 |
| 60° | √3/2 ≈ 0.866 | 0.5 | √3 ≈ 1.732 |
| 90° | 1 | 0 | undefined |
Notice the symmetry: sin(30°) = cos(60°) = 0.5. Sine of an angle always equals cosine of its complement (90° − angle). That pattern halves what you need to memorise.
Worked Example 1: The Ladder
A 5 m ladder leans against a wall, making a 60° angle with the ground. How high up the wall does it reach?
The ladder is the hypotenuse (5 m). The height is the side opposite the 60° angle. Opposite + hypotenuse → sine:
Height = 5 × sin(60°) = 5 × 0.866 = 4.33 m
Worked Example 2: The Building
You stand 20 m from a building. The angle from your eyes to the top is 35°. How tall is the building? (Assume eye level at 1.7 m.)
Distance is adjacent, height above eye level is opposite → tangent:
Height above eyes = 20 × tan(35°) = 20 × 0.700 = 14.0 m
Total height = 14.0 + 1.7 = 15.7 m
This is exactly how surveyors measured building heights for centuries: one angle, one distance, no climbing.
Worked Example 3: Going Backwards (Inverse Functions)
A ramp rises 1.2 m over a horizontal run of 6 m. What's its angle? Here you know opposite and adjacent, so tan(θ) = 1.2 ÷ 6 = 0.2. To get θ itself, use the inverse tangent:
θ = arctan(0.2) ≈ 11.3°
Inverse functions (arcsin, arccos, arctan, sometimes written sin⁻¹, cos⁻¹, tan⁻¹) convert a ratio back into an angle. They're how you answer "what angle gives this slope?"
Degrees vs Radians
Calculators have a DEG/RAD mode and using the wrong one is the classic trig error. Degrees split a circle into 360 parts; radians measure angle by arc length, with a full circle = 2π ≈ 6.283 radians. School problems use degrees; calculus, physics and programming use radians. Always check the mode before trusting an answer: sin(30) in radian mode gives −0.988, not 0.5.
Where This Actually Gets Used
- Construction: roof pitches, stair angles, cutting mitres
- Navigation: GPS triangulation, bearing calculations
- Physics/engineering: resolving forces into components, wave analysis
- Game development: aiming, projectile arcs, camera rotation
- Electrical: AC phase angles, three-phase power
Try Our Calculators
Use the trigonometry calculator for all six trig functions of any angle, with a unit circle diagram. To solve a complete triangle from any two known values, the right triangle solver finds every side, angle and the area. For longer calculations the scientific calculator has sin, cos and tan in both degrees and radians, with a running history.