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Tangent Calculator
Use this tangent calculator to find tan of any angle, or work backwards with arctan (inverse tangent) to find the angle from a ratio. Switch between degrees and radians, and watch the unit circle update as you type so you can see exactly where the angle sits.
Tan is undefined at 90°, 270°, and every 180° after that.
For tan, enter the angle. For arctan, enter any number and the answer is the angle whose tangent is that number.
How it works
Tangent is one of the three basic trigonometric ratios, alongside sine and cosine. In a right triangle, tan of an angle is the length of the side opposite the angle divided by the length of the side next to it (the adjacent side). If the opposite side is 3 and the adjacent side is 4, tan of that angle is 3 ÷ 4 = 0.75. The same ratio appears on the unit circle, where the radius is 1: the horizontal run of the radius is cos, the vertical rise is sin, and tan is sin divided by cos.
Degrees and radians are two ways of measuring the same angles. A full circle is 360 degrees, or 2π radians (about 6.2832). To convert degrees to radians, multiply by π ÷ 180; to convert radians to degrees, multiply by 180 ÷ π. So 45 degrees is π ÷ 4 radians, about 0.7854, and 1 radian is about 57.2958 degrees. School maths and everyday work usually use degrees, while calculus and physics often use radians — this calculator lets you use either, and the answer is given in the same unit you chose.
Arctan, or inverse tangent, runs the process backwards. Tan takes an angle and gives a ratio; arctan takes a ratio and gives the angle. Arctan(1) is 45 degrees, because tan(45°) = 1. Arctan accepts any number at all — positive, negative, or zero — and always returns an angle between −90° and 90°. That range matters: tan repeats every 180 degrees, so many angles share the same tangent, and arctan picks the one between −90° and 90°. If you need an angle in another quadrant, add or subtract 180 degrees from the result.
Where tan is undefined is worth knowing before you trust an answer. Tan is sin divided by cos, so whenever cos is zero, tan has no value — you would be dividing by zero. That happens at 90°, 270°, and every 180° after that. Just short of those angles, tan is enormous: tan(89°) is about 57.29, and tan(89.9°) is about 572.96. This calculator shows a friendly message instead of a meaningless number when you enter one of those angles.
Common values are a good way to check your answers. Tan(0°) = 0, tan(30°) ≈ 0.5774, tan(45°) = 1, tan(60°) ≈ 1.7321, and tan(90°) is undefined. On the negative side, tan(−45°) = −1. Between 0° and 90°, tan grows from 0 to very large, which is why it is so useful for slopes: a small change in angle near 90° makes a huge difference to the ratio, just as a small change in a steep hill's angle makes a big difference to how hard it is to climb.
Real uses turn up wherever a slope has to become an angle. A roof with a rise of 1 for every 4 of run has a pitch of arctan(1 ÷ 4) ≈ 14.04°. A surveyor who stands 100 feet from a building and measures the angle to the top as 30° finds the height is 100 × tan(30°) ≈ 57.74 feet. Road signs give gradients as ratios — a 1 in 10 slope is arctan(0.1) ≈ 5.71°. In physics, tan gives the direction of a velocity or a force from its horizontal and vertical parts.
Worked example
Find tan(45°). This is the simplest tangent problem, and it shows the whole idea in one step.
Step 1: Picture a right triangle with an angle of 45°. The other acute angle is also 45°, so the two legs are equal in length.
Step 2: Say each leg is 1 unit long. The opposite side is 1 and the adjacent side is 1.
Step 3: Tan is opposite divided by adjacent, so tan(45°) = 1 ÷ 1 = 1.
Step 4: Check on the unit circle. Sin(45°) and cos(45°) are both about 0.7071, and 0.7071 ÷ 0.7071 = 1.
Try it yourself: choose "Tangent tan(x)", leave the unit on Degrees, enter 45, and press Calculate. The tool gives exactly the same answer — tan(45°) = 1. Then switch to "Inverse tangent arctan(x)", enter 1, and you get 45° back, confirming that tan and arctan undo each other.
FAQ
What is tan of an angle?
Tan of an angle is the ratio of the opposite side to the adjacent side in a right triangle. It is also equal to sin divided by cos. For example, tan(45 degrees) equals 1, because the opposite and adjacent sides are equal in a 45-degree right triangle. Tan grows quickly as the angle approaches 90 degrees, where it is undefined.
How do I switch between degrees and radians?
Use the Degrees / Radians selector above the angle box. In degrees, a full circle is 360 degrees; in radians, a full circle is 2 pi, about 6.2832 radians. To convert, multiply degrees by pi divided by 180 to get radians, or multiply radians by 180 divided by pi to get degrees. For example, 45 degrees equals pi divided by 4, which is about 0.7854 radians.
What is arctan (inverse tangent)?
Arctan is the reverse of tan: it takes a ratio and returns the angle that has that tangent. Arctan(1) is 45 degrees, because tan(45 degrees) equals 1. Arctan accepts any number, positive or negative, and always returns an angle between -90 and 90 degrees. It is the tool to reach for when you know a slope or a ratio and need the angle behind it.
Why is tan(90 degrees) undefined?
Tan is sin divided by cos, and cos(90 degrees) is 0, so tan(90 degrees) would mean dividing by zero, which has no answer. As the angle gets closer to 90 degrees, tan gets larger and larger — tan(89 degrees) is about 57.29, tan(89.9 degrees) is about 572.96 — but it never settles on a value at 90 degrees itself. The same happens at 270 degrees, and at every angle 180 degrees further on.
What is tan used for in real life?
Tan is used whenever a slope or a gradient has to become an angle. Roofers use it to turn a roof's rise and run into a pitch angle, surveyors use it to find the height of a building from a measured distance and an angle, and road engineers use it for the grade of a hill. A slope of 1 in 10, a rise of 1 for every 10 of run, is an angle of arctan(0.1), about 5.71 degrees. In physics, tan describes the direction of a velocity or a force.