Tangent Calculator
Calculate tan(x) for any angle or find the angle from a tangent value using arctan. Features unit circle and tangent graph with asymptotes.
tan(x) = opposite / adjacent = sin(x) / cos(x)Result
tan(45)
1.000000
Exact: 1
Calculation Mode
Input
Enter angle in degrees
Unit Circle Visualization
On the unit circle, tan(45.0) = sin / cos = 0.707107 / 0.707107 = 1.000000
Related Trigonometric Values
tan(45)
1.000000
sin(45)
0.707107
cos(45)
0.707107
cot(45) = 1/tan
1.000000
Slope Interpretation: A line at 45 from horizontal has slope = tan(45) = 1.000000
Tangent Wave Graph
Amplitude
1
Period
2pi
Phase Shift
0
Vertical Shift
0
Equation:
y = tan((x))
Vertical Asymptotes
Tangent is undefined where cosine equals zero. These occur at odd multiples of 90 degrees:
-270
tan undefined
-90
tan undefined
90
tan undefined
270
tan undefined
General formula: x = 90 + n*180 (or pi/2 + n*pi radians), where n is any integer.
Special Tangent Values
| Angle (deg) | Radians | tan | Exact Value |
|---|---|---|---|
| 0 | 0 | 0.0000 | 0 |
| 30 | pi/6 | 0.5774 | sqrt(3)/3 |
| 45 | pi/4 | 1.0000 | 1 |
| 60 | pi/3 | 1.7321 | sqrt(3) |
| 90 | pi/2 | undef | undefined |
| 120 | 2pi/3 | -1.7321 | -sqrt(3) |
| 135 | 3pi/4 | -1.0000 | -1 |
| 150 | 5pi/6 | -0.5774 | -sqrt(3)/3 |
| 180 | pi | 0.0000 | 0 |
Tangent Function Properties
Domain
All reals except x = 90 + n*180
Range
All real numbers (-inf, inf)
Period
180 degrees (pi radians)
Odd/Even
Odd function: tan(-x) = -tan(x)
Zeros
x = n*180, where n is integer
Asymptotes
x = 90 + n*180, where n is integer
Applications of Tangent
Slope Calculation
The tangent of an angle equals the slope of a line. If road has 15% grade, angle = arctan(0.15) ≈ 8.5.
Height Estimation
Height = distance * tan(elevation angle). Used in surveying and navigation.
Engineering
Calculating roof pitch, ramp grades, and angles of inclination in construction.
Physics
Projectile motion (launch angle), friction on inclined planes, and optics.
Result
tan(45)
1.000000
?What is Tangent?
Tangent (tan) is a trigonometric function that gives the ratio of the opposite side to the adjacent side in a right triangle: tan(angle) = opposite/adjacent = sin/cos. Unlike sine and cosine, tangent has no bounds - it ranges from negative infinity to positive infinity. Tangent is undefined at 90 and 270 degrees (vertical asymptotes). Common values: tan(0)=0, tan(30)=sqrt(3)/3, tan(45)=1, tan(60)=sqrt(3).
About the Tangent Function
The tangent function is the ratio of sine to cosine: tan(x) = sin(x)/cos(x). In a right triangle, tangent equals opposite over adjacent. Unlike sine and cosine, tangent is unbounded and has vertical asymptotes where cosine equals zero. Tangent is especially important in calculus as the slope of curves, in physics for angles of inclination, and in engineering for grade calculations.
Key Facts
- tan(x) = opposite / adjacent = sin(x) / cos(x)
- Range: all real numbers (-infinity to +infinity)
- Period: 180 degrees (pi radians) - half of sine/cosine
- Undefined at 90, 270 (asymptotes where cos = 0)
- tan(0) = 0, tan(45) = 1, tan(90) = undefined
- arctan(x) returns angle in range (-90, 90) or (-pi/2, pi/2)
- Tangent is positive in quadrants I and III
- tan(x) represents slope in geometry: rise/run
Frequently Asked Questions
Tangent is a trigonometric function defined as the ratio of sine to cosine: tan(x) = sin(x)/cos(x). In a right triangle, tan(angle) = opposite/adjacent. Unlike sine and cosine, tangent can take any real value from negative to positive infinity.
Since tan = sin/cos, and cos(90) = 0, we have tan(90) = sin(90)/0 = 1/0, which is undefined. These points (90, 270, etc.) are called vertical asymptotes where the function approaches positive or negative infinity.
Arctan (also written as tan inverse or atan) returns the angle whose tangent equals a given value. For example, arctan(1) = 45 because tan(45) = 1. The result is always between -90 and 90 degrees (exclusive).
Tangent equals rise over run (opposite/adjacent), which is exactly the definition of slope. If a line makes an angle x with the positive x-axis, its slope m = tan(x). A 45-degree line has slope 1 because tan(45) = 1.
The tangent function has a period of 180 degrees (pi radians), which is half the period of sine and cosine. This means tan(x) = tan(x + 180) for any angle x. This is because tangent repeats its values more frequently.
Last updated: 2025-01-15
Result
tan(45)
1.000000
Exact: 1