Trigonometry Calculator
What is Trigonometry?
Trigonometry (from Greek trigonon "triangle" + metron "measure") is the branch of math that studies the relationship between side lengths and angles of triangles.
While algebra solves equations for $x$, trigonometry allows us to calculate distances and heights that are impossible to measure with a tape measure—like the height of a mountain or the distance to a star.
The Magic Mnemonic: SOH CAH TOA
The entire foundation of Right Triangle math rests on three ratios. To memorize them, students use the acronym SOH CAH TOA.
1. Sine (SOH)
2. Cosine (CAH)
3. Tangent (TOA)
The Pythagorean Connection
If you only know two sides but no angles (other than the 90° right angle), you don't need trig functions. You can use the Pythagorean Theorem.
$a^2 + b^2 = c^2$
Our calculator automatically switches between these methods. If you enter two sides, it uses Pythagoras. If you enter an angle, it uses SOH CAH TOA.
Inverse Functions (Finding the Angle)
What if you know the sides but need to find the angle? You use the "Arc" functions (Inverse Trig).
- Arcsin ($sin^{-1}$): Undoes Sine.
- Arccos ($cos^{-1}$): Undoes Cosine.
- Arctan ($tan^{-1}$): Undoes Tangent.
If Opposite = 3 and Adjacent = 4.
$Tan(\theta) = 3/4 = 0.75$.
$\theta = arctan(0.75) = 36.87^{\circ}$.
Real World Uses
Trigonometry is the hidden math behind much of the modern world.
| Industry | Application |
|---|---|
| Video Games | Calculating camera angles, object physics, and rendering 3D graphics on a 2D screen. |
| Construction | Roof pitches, ramp slopes, and structural integrity of bridges (Trusses). |
| Aviation | Calculating flight paths to account for wind speed vectors. |
| Astronomy | Using parallax (triangle geometry) to measure the distance to nearby stars. |