Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is the branch of mathematics that studies relationships between side lengths and angles of triangles. While it starts with simple right-angled triangles, it is the foundation for modeling waves, sound, light, and analyzing periodic phenomena.
It is the mathematical bridge that allows us to calculate distances we cannot measure directly, such as the height of a mountain or the distance to a star.
1. The Right-Angled Triangle
Trigonometry begins with the right-angled triangle. To understand the functions, we must first name the three sides relative to a specific angle, usually denoted by the Greek letter Theta (θ).
[Image of right triangle with hypotenuse opposite and adjacent labeled]- Hypotenuse: The longest side, opposite the right angle (90°).
- Opposite: The side directly across from the angle θ.
- Adjacent: The side next to the angle θ (that isn't the hypotenuse).
2. The Three Primary Ratios (SOH CAH TOA)
The core of trigonometry relies on three ratios that compare lengths of the sides. We use the mnemonic SOH CAH TOA to remember them.
[Image of SOH CAH TOA mnemonic diagram]Sine (sin)
Cosine (cos)
Tangent (tan)
3. Example Calculation
Imagine a triangle where the Hypotenuse is 10cm, and the angle θ is 30°. How long is the Opposite side?
Using SOH (Sine = Opposite / Hypotenuse):
0.5 = x / 10
x = 10 × 0.5 = 5 cm
4. Common Trigonometric Values
There are specific angles that appear frequently in mathematics. Memorizing these values is very helpful.
| Angle (θ) | sin(θ) | cos(θ) | tan(θ) |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | Undefined |
5. The Unit Circle
Trigonometry isn't limited to triangles with angles less than 90°. To calculate values for angles like 120° or 300°, we use the Unit Circle—a circle with a radius of 1 centered at the origin.
[Image of unit circle trigonometry]On the unit circle, the x-coordinate of a point represents the cosine, and the y-coordinate represents the sine: (x, y) = (cos θ, sin θ).
6. The Pythagorean Identity
Because the Unit Circle relates to a right triangle with a hypotenuse of 1, applying the Pythagorean theorem (a² + b² = c²) gives us the most famous identity in trigonometry:
Conclusion
Trigonometry is a vast field. While SOH CAH TOA helps us measure triangles, the sine and cosine functions (waves) help us understand the universe, from the orbit of planets to the music on your phone.