Tangent calculator
Enter an angle and get its tangent straight away, with the exact value for special angles such as tan(45°) = 1.
In degrees or radians (choose above).
Fill in the fields; the answer appears here straight away.
How it's worked out
How it works
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Enter the values
Type the numbers or the formula. Decimals may use a point or a comma. No idea? Click Fill in an example.
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Instant answer
The answer appears as you type, with the most important intermediate values.
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See the working
Under "How it's worked out" you see the steps, handy for checking your own calculation.
What is the tangent?
In a right-angled triangle, the tangent of an angle is the opposite side divided by the adjacent side. It is also sine divided by cosine. That is why the tangent doesn't exist at 90° and 270°: the cosine is 0 there, and the calculator tells you so.
Degrees or radians
Enter the angle in degrees or radians (choose above). Expressions such as pi/3 or 2pi/5 work too.
Useful values
tan(0°) = 0, tan(30°) = √3/3 ≈ 0.577, tan(45°) = 1 and tan(60°) = √3 ≈ 1.732. To find an angle from its tangent, use the arctan calculator. For a triangle with two known sides, use the right triangle calculator.
Values that come up often
| Angle | tan | Decimal |
|---|---|---|
| 0° | 0 | 0 |
| 30° | √3/3 | 0.5774 |
| 45° | 1 | 1 |
| 60° | √3 | 1.7321 |
| 89° | 57.29 |
Notice how fast the tangent grows near 90°: the closer you get, the larger it becomes without limit.
Height from a shadow
A pole casts a shadow of 12 m while the sun stands 35° above the horizon. The height is 12 × tan(35°) ≈ 12 × 0.7002 ≈ 8.4 m. The same idea measures the height of a tree or building from a known distance and the angle you look up at the top.
Slope and gradient
A road with a 12% gradient rises 12 m per 100 m, so the angle is arctan(0.12) ≈ 6.8°. Use the arctan calculator for that, and the trigonometry calculator to see sine, cosine and tangent for one angle at once.
Frequently asked questions
Why is tan 90° undefined?
Because tan = sin ÷ cos, and cos(90°) = 0. Dividing by zero has no answer, so the tangent does not exist at 90° or 270°, and the calculator tells you.
What does the tangent mean for a slope?
It is the gradient: rise ÷ run. A slope of 10% has tan(angle) = 0.1, which is an angle of about 5.7°. To go from the gradient to the angle, use arctan.
Can the tangent be negative?
Yes, for angles between 90° and 180° and between 270° and 360°. tan(135°) = −1.
How often does the tangent repeat?
Every 180° (π radians), twice as fast as the sine and cosine. tan(45°) = tan(225°) = 1.