Tangent Calculator
Calculate tangent (tan) values for angles or find angles from tangent values
Calculate Tangent Values
Enter the angle to calculate its tangent value
Tangent Result
Calculation Breakdown:
Formula: tan(θ) = sin(θ) / cos(θ)
sin(45°): 0.70710678
cos(45°): 0.70710678
tan(45°): 0.70710678 / 0.70710678 = 1
Common Angle Examples
Tangent Reference Table
| Angle (Degrees) | Angle (Radians) | Exact Value | Decimal |
|---|---|---|---|
| 0° | 0 | 0 | 0 |
| 30° | π/6 | √3/3 | 0.577 |
| 45° | π/4 | 1 | 1 |
| 60° | π/3 | √3 | 1.732 |
| 90° | π/2 | Undefined | Undefined |
| 180° | π | 0 | 0 |
| 270° | 3π/2 | Undefined | Undefined |
Tangent Properties
Definition
tan(θ) = opposite / adjacent = sin(θ) / cos(θ)
Domain
All real numbers except odd multiples of π/2
Range
All real numbers (-∞, +∞)
Period
π radians (180°)
Tangent Tips
Tangent is undefined when cosine equals zero
tan(θ) = tan(θ + 180°) - tangent has period 180°
tan(45°) = 1 is the most commonly used value
Negative angles: tan(-θ) = -tan(θ)
Used in slopes, angles, and right triangle calculations
Understanding the Tangent Function
What is Tangent?
The tangent function is one of the fundamental trigonometric functions. In a right triangle, tangent is the ratio of the opposite side to the adjacent side. On the unit circle, tangent represents the slope of the line from the origin to a point on the circle.
Key Applications
- •Calculating slopes and angles in geometry
- •Engineering and physics calculations
- •Navigation and surveying
- •Architecture and construction
Mathematical Properties
Basic Formula
tan(θ) = sin(θ) / cos(θ)
Right Triangle
tan(θ) = opposite / adjacent
Inverse Function
θ = arctan(x) = tan⁻¹(x)
Law of Tangents
The law of tangents relates the tangent of two angles of a triangle to the lengths of the opposite sides:
Where a and b are side lengths, and α and β are the opposite angles.
Common Mistakes to Avoid
• Remember that tangent is undefined at 90°, 270°, etc. (when cos(θ) = 0)
• Don't confuse tangent with the geometric concept of a tangent line
• Be careful with angle units - always specify degrees or radians
• Remember that tangent has a period of 180° (π radians), not 360°