Calculate the tangent of any angle, find an angle from its tangent with arctan, or get tan from right-triangle sides in degrees or radians.
Tan Formula
The tangent of an angle in a right triangle is the ratio of the side opposite the angle to the side next to it. It is also equal to sine divided by cosine.
tan(θ) = opposite / adjacent
tan(θ) = sin(θ) / cos(θ)
To go the other way and find the angle from a tangent value, use the inverse tangent (arctan):
θ = arctan(x)
- θ (theta) is the angle, measured in degrees or radians.
- opposite is the length of the side across from the angle.
- adjacent is the length of the side touching the angle (not the hypotenuse).
- sin(θ) is the sine of the angle.
- cos(θ) is the cosine of the angle.
- x is a tangent value, equal to opposite divided by adjacent.
The calculator has three modes. The first takes an angle and returns its tangent, so you enter θ and read off tan(θ). The second takes a tangent value and returns the angle that produces it using arctan. The third takes the opposite and adjacent sides of a right triangle and returns both the tangent ratio and the angle. Switch the angle unit between degrees and radians to match your problem, and turn on the options to set decimal places or to also show sine, cosine, cotangent, secant, and cosecant for the same angle.
Common Tangent Values
These reference values cover the angles you meet most often. Tangent repeats every 180 degrees and is undefined at 90 and 270 degrees because the cosine there is zero.
| Angle (degrees) | Angle (radians) | tan(θ) |
|---|---|---|
| 0 | 0 | 0 |
| 30 | π/6 | 0.5774 |
| 45 | π/4 | 1 |
| 60 | π/3 | 1.7321 |
| 90 | π/2 | Undefined |
| 120 | 2π/3 | -1.7321 |
| 135 | 3π/4 | -1 |
| 180 | π | 0 |
When you read a result, the sign tells you the quadrant behavior: tangent is positive in the first and third quadrants and negative in the second and fourth. A tangent of 1 means the opposite and adjacent sides are equal, which is the 45 degree case.
Example Problems
Example 1. Find the tangent of a 30 degree angle. Set the mode to tangent of an angle, enter 30, and keep the unit on degrees. The calculator returns tan(30) = 0.5774. This matches the reference table above.
Example 2. A right triangle has an opposite side of 4 and an adjacent side of 3. Set the mode to tangent from sides and enter 4 and 3. The tangent ratio is 4 / 3 = 1.3333, and the angle is arctan(1.3333) = 53.13 degrees.
Frequently Asked Questions
Why is tan(90) undefined? Tangent equals sine divided by cosine, and the cosine of 90 degrees is zero. Dividing by zero has no defined result, so the tangent is undefined at 90 degrees and at every angle 180 degrees away from it, such as 270 degrees. Near those angles the value grows without bound, so a calculator may show a very large number for an angle close to 90.
What is the difference between tan and arctan? Tangent takes an angle and gives you a ratio. Arctan, the inverse tangent, takes a ratio and gives you back the angle. Use tangent when you know the angle and want the side ratio, and use arctan when you know the sides or the ratio and want the angle.
Should I use degrees or radians? Use whichever unit your problem is stated in. Geometry and everyday angle problems usually use degrees, while calculus and physics often use radians. The calculator lets you pick the unit so you do not have to convert by hand, but make sure the selector matches the value you typed in.
