Six Trigonometric Functions- Complete Guide
What Are the Six Trigonometric Functions?
Trigonometry centers on six core functions. These aren't abstract concepts designed to torture students — they're tools for measuring angles and relationships in triangles.
The six trigonometric functions are:
- Sine (sin)
- Cosine (cos)
- Tangent (tan)
- Cosecant (csc)
- Secant (sec)
- Cotangent (cot)
Each function takes an angle as input and outputs a ratio. That's the whole game.
The Primary Three: Sine, Cosine, and Tangent
These three form the foundation. You need to know these cold before touching anything else.
Sine (sin)
In a right triangle, sine equals the opposite side divided by the hypotenuse.
sin(θ) = opposite / hypotenuse
Cosine (cos)
Cosine equals the adjacent side divided by the hypotenuse.
cos(θ) = adjacent / hypotenuse
Tangent (tan)
Tangent equals opposite divided by adjacent. You can also calculate it as sin(θ) / cos(θ).
tan(θ) = opposite / adjacent
Here's a quick way to remember: SOH CAH TOA. Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.
The Reciprocal Three: Cosecant, Secant, and Cotangent
These are just inverses of the primary three. Nothing fancy.
- Cosecant (csc) = 1 / sin(θ) = hypotenuse / opposite
- Secant (sec) = 1 / cos(θ) = hypotenuse / adjacent
- Cotangent (cot) = 1 / tan(θ) = adjacent / opposite
Most problems stick to sine, cosine, and tangent. You'll encounter the reciprocals in calculus and when working with identities.
Understanding the Unit Circle
The unit circle extends trigonometry beyond right triangles. It has a radius of 1, centered at the origin.
On this circle:
- The x-coordinate of any point equals cos(θ)
- The y-coordinate of any point equals sin(θ)
- tan(θ) = y/x = sin(θ)/cos(θ)
This matters because it lets you evaluate trig functions for any angle — not just acute angles in triangles.
Six Trigonometric Functions at a Glance
| Function | Abbreviation | Definition | Reciprocal |
|---|---|---|---|
| Sine | sin | opposite ÷ hypotenuse | Cosecant |
| Cosine | cos | adjacent ÷ hypotenuse | Secant |
| Tangent | tan | opposite ÷ adjacent | Cotangent |
| Cosecant | csc | hypotenuse ÷ opposite | Sine |
| Secant | sec | hypotenuse ÷ adjacent | Cosine |
| Cotangent | cot | adjacent ÷ opposite | Tangent |
How to Calculate Trigonometric Functions
Using a Calculator
Most scientific calculators have sin, cos, and tan buttons. Make sure your calculator is in DEG mode for degrees or RAD mode for radians, depending on what your problem uses.
For csc, sec, and cot, you'll need to calculate the reciprocal yourself. Find sin/cos/tan first, then press the 1/x button or divide 1 by your result.
Using Common Angle Values
Memorize these values — they come up constantly:
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
Step-by-Step Example
Find sin(45°), cos(45°), and tan(45°) for a right triangle where both legs equal 5.
- Identify sides: both legs are 5, so opposite = adjacent = 5
- Find hypotenuse: √(5² + 5²) = √50 = 5√2
- Calculate sin(45°): opposite/hypotenuse = 5/(5√2) = 1/√2 = √2/2 ✓
- Calculate cos(45°): adjacent/hypotenuse = 5/(5√2) = √2/2 ✓
- Calculate tan(45°): opposite/adjacent = 5/5 = 1 ✓
Where These Functions Actually Show Up
You won't just see these in textbooks. Real applications exist:
- Architecture — calculating roof slopes, structural loads, and support angles
- Physics — analyzing wave patterns, projectile motion, and forces at angles
- Engineering — signal processing, alternating current calculations, mechanical systems
- Computer graphics — rotations, lighting calculations, and 3D rendering
- Navigation — GPS systems and map projections
Common Mistakes That Cost Points
- Confusing adjacent and opposite — always label your triangle before touching any formula
- Forgetting to check your mode — degrees vs radians will give completely wrong answers
- Using the wrong function — if a problem asks for tan and you calculate sin, you've failed the question
- Leaving answers unsimplified — √2/2 is cleaner than 0.707
- Dividing when you should multiply — reciprocal doesn't mean flip everything, it means 1 divided by the original value
The Bottom Line
The six trigonometric functions boil down to three ratios and their inverses. Learn SOH CAH TOA first. Master the common angle values. Understand that csc, sec, and cot exist only because mathematicians enjoy having options.
Once you know how to set up a right triangle and identify which sides go where, any trig problem becomes straightforward. Stop memorizing everything — understand the relationships instead.