All Six Trigonometric Functions: Sin, Cos, Tan, Csc, Sec, Cot Explained
Understand all six trigonometric functions in one guide. Learn sin, cos, tan, csc, sec, and cot with formulas, values, and how they relate to each other.
Trigonometry is built on six fundamental functions: sine, cosine, tangent, cosecant, secant, and cotangent. Each function describes a different relationship between the angles and sides of a right triangle. Understanding all six — and how they relate to each other — is essential for solving triangles, analyzing waves, and working with the unit circle.
Key Takeaway
The three primary functions (sin, cos, tan) and three reciprocal functions (csc, sec, cot) together describe all possible ratio relationships in a right triangle. Each function is undefined at specific angles where its denominator equals zero.
1. The Three Primary Functions
| Function | Formula | Unit Circle |
|---|---|---|
| sin(θ) | opposite / hypotenuse | y-coordinate |
| cos(θ) | adjacent / hypotenuse | x-coordinate |
| tan(θ) | opposite / adjacent | y/x = slope |
2. The Three Reciprocal Functions
| Function | Reciprocal Of | Undefined When |
|---|---|---|
| csc(θ) | 1/sin(θ) | sin(θ) = 0 |
| sec(θ) | 1/cos(θ) | cos(θ) = 0 |
| cot(θ) | 1/tan(θ) = cos/sin | sin(θ) = 0 |
3. Signs by Quadrant
- Quadrant I (0°-90°): All six functions are positive.
- Quadrant II (90°-180°): Only sin and csc are positive.
- Quadrant III (180°-270°): Only tan and cot are positive.
- Quadrant IV (270°-360°): Only cos and sec are positive.
4. Frequently Asked Questions
Which trig functions can be greater than 1?
Tan, cot, sec, and csc can all exceed 1 in absolute value. Only sin and cos are restricted to [-1, 1].
How do I find all six functions from one known value?
Use Pythagorean identities and the quadrant information. If you know sin(θ) = 0.6 in Q1, then cos(θ) = 0.8, and you can derive all others.
Try Our Calculator
Use our free Trigonometric Functions Calculator to see all six values for any angle at once.