Tangent Calculator

The tangent function, tan(x)

The tangent of an angle compares how steeply a line rises with how far it runs. In a right triangle it is the length of the side opposite the angle divided by the length of the side next to it (the adjacent side). The hypotenuse plays no part.

tan(x) = opposite ÷ adjacent = sin(x) ÷ cos(x)The second form works for any angle, not only those inside a right triangle.

Geometrically, tan(x) is the slope of a line that makes an angle x with the horizontal axis. A flat line has tan(0) = 0, a line at 45° has slope 1, and the line gets steeper without limit as the angle approaches 90°, where the tangent is not defined.

• Domain: every angle except 90° + k·180° (π/2 + kπ in radians), where cos(x) = 0.
• Range: all real numbers, with no upper or lower bound.
• Period: 180° (π radians), so tan(x + 180°) = tan(x).
• Symmetry: tan(−x) = −tan(x), which makes it an odd function.

Tangent values worth remembering

A few angles have tangents that can be written exactly. Everything else is a decimal approximation, which is what the calculator returns (rounded to six places).

Notice how fast the values grow near 90°: moving from 85° to 89° multiplies the tangent by five.

How to calculate the tangent of an angle

1. Enter the angle in the Angle box.
2. Choose degrees or radians from the drop-down. Using the wrong unit is the most common source of wrong answers.
3. Press Calculate. The result is shown as tan(angle) = value, rounded to at most six decimal places.

Example 1, by ratio. A right triangle has an angle of 35°. Its sine is 0.573576 and its cosine is 0.819152, so tan(35°) = 0.573576 ÷ 0.819152 = 0.700208.

Example 2, radians. For x = 1 rad the calculator gives tan(1) = 1.557408. Converting first would give the same answer: 1 rad ≈ 57.2958°, and tan(57.2958°) is about 1.5574.

radians = degrees × π ÷ 180Use this to move between the two angle units by hand.

At 90° (and 270°, 450° and so on) the calculator reports that the tangent is not defined. Close to those angles the tangent is a very large number, for instance tan(89.9999°) = 572,957.795104.

Related trigonometric functions

The tangent is tied to the other trigonometric functions by a handful of identities that are useful for checking a result or simplifying an expression.

• Cotangent, the reciprocal: cot(x) = 1 ÷ tan(x). See the cotangent calculator.
• Pythagorean identity: 1 + tan²(x) = sec²(x), where sec(x) = 1 ÷ cos(x).
• Inverse functions: arctan(tan x) returns the original angle (within −90° to 90°). The arctan calculator goes from a ratio back to an angle, and the arccotangent calculator does the same for the cotangent.
• Sine and cosine: compare with the sine calculator and the cosine calculator, or use the trigonometry calculator to switch between all twelve functions.

The symmetry of the unit circle gives further shortcuts, such as tan(180° − x) = −tan(x). For example tan(100°) = −tan(80°) = −5.671282.

Where the tangent is used

Whenever a problem links a vertical change to a horizontal distance through an angle, the tangent is probably involved.

• Heights and distances. Standing 50 m from a tree and measuring an angle of elevation of 32°, the height above eye level is 50 × tan(32°) = 31.24 m.
• Ladders and ramps. A ladder that makes 70° with the ground and whose foot is 2 m from the wall reaches 2 × tan(70°) = 5.49 m up the wall.
• Road and roof grades. A slope of 5° has a gradient of tan(5°) × 100 = 8.75%. A 10% grade is about 5.7°.
• Navigation and surveying. Bearings and offsets are converted to coordinates with tangents, and the reverse step uses the arctangent.
• Physics and engineering. The angle of friction, the phase angle in AC circuits and the direction of a force vector are all given by a tangent of a ratio.

To find an angle from two sides instead, divide opposite by adjacent and apply the inverse function. If the wall is 18 m high and you stand 40 m away, the elevation angle is arctan(18 ÷ 40) = arctan(0.45) ≈ 24.23°.

Common mistakes

• Leaving a calculator in radian mode when the angle is in degrees, or the other way round. tan(45) is 1 in degrees but about 1.6198 in radians.
• Treating the tangent as bounded. Unlike sine and cosine, it can be larger than 1 or smaller than −1.
• Expecting a value at 90°. The function has a vertical asymptote there, not a very large finite number.
• Rounding too early. If the tangent feeds into a later step, keep all the digits until the end.