tan(x), tangent function.
Tangent definition
In a right triangle ABC the tangent of α, tan(α) is defined as the ratio between the side opposite to angle α and the side adjacent to the angle α:
tan α = a / b
Example
a = 3"
b = 4"
tan α = a / b = 3 / 4 = 0.75
Graph of tangent
Tangent rules
Rule name | Rule |
---|---|
Symmetry | tan(-θ) = -tan θ |
Symmetry | tan(90°- θ) = cot θ |
tan θ = sin θ / cos θ | |
tan θ = 1 / cot θ | |
Double angle | tan 2θ = 2 tan θ / (1 - tan^{2} θ) |
Angles sum | tan(α+β) = (tan α + tan β) / (1 - tan α tan β) |
Angles difference | tan(α-β) = (tan α - tan β) / (1 + tan α tan β) |
Derivative | tan' x = 1 / cos^{2}(x) |
Integral | ∫ tan x dx = - ln |cos x| + C |
Euler's formula | tan x = (e^{ix} - e^{-ix}) / i (e^{ix} + e^{-ix}) |
Inverse tangent function
The arctangent of x is defined as the inverse tangent function of x when x is real (x∈ℝ).
When the tangent of y is equal to x:
tan y = x
Then the arctangent of x is equal to the inverse tangent function of x, which is equal to y:
arctan x = tan^{-1} x = y
Example
arctan 1 = tan^{-1} 1 = π/4 rad = 45°
See: Arctan function
Tangent table
x (rad) |
x (°) |
tan(x) |
---|---|---|
-π/2 | -90° | -∞ |
-1.2490 | -71.565° | -3 |
-1.1071 | -63.435° | -2 |
-π/3 | -60° | -√3 |
-π/4 | -45° | -1 |
-π/6 | -30° | -1/√3 |
-0.4636 | -26.565° | -0.5 |
0 | 0° | 0 |
0.4636 | 26.565° | 0.5 |
π/6 | 30° | 1/√3 |
π/4 | 45° | 1 |
π/3 | 60° | √3 |
1.1071 | 63.435° | 2 |
1.2490 | 71.565° | 3 |
π/2 | 90° | ∞ |
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