Tangent Calculator

Welcome to the Tangent Calculator, which is used to calculate the tangent of degrees or radians. Arbitrary degrees or radians can be entered, such as integers, decimals, or fractions. In addition, radian π combinations are also supported.

In a right triangle, the ratio of the opposite side to the adjacent side of an acute angle is called the tangent. Abbreviated as tan.

The tangent formula is

tan(θ) = oppositeadjacent

For example, there is a right triangle as follows

In this right triangle, the three angles are α, β and γ, where γ is the right angle. The three sides are a, b, and c, where c is the hypotenuse. Therefore, the tangents of angles α and β are

tan(α) = oppositeadjacent = ab

 

tan(β) = adjacentopposite = ba

It can be seen that the tangents of these two angles are reciprocals of each other.

Another angle γ is a right angle, so what is the tangent of the right angle?

Like cosine, when there are two right angles in a right triangle (90°, 90°, 0), then the sides adjacent to the right angles will tend to 0. And the denominator of the fraction must not be zero, so the tangent of the right angle does not exist, or it is infinite.

In a right triangle, calculating the tangent of an acute angle is easy when the lengths of the three sides are known.

For example, in a right triangle, the lengths of the three sides are 6, 8, and 10 respectively, as shown in the figure below, so what is the tangent of the angle α?

It can be seen that the sides opposite and adjacent to angle α are 6 and 8. Therefore, the tangent of angle α is

tan(α) = 68 = 34

In addition, if the length of the side is unknown but the degree of the angle is known. Then, its tangent can be calculated with the help of a calculator. Like the tangent calculator available on this page. Also, you can refer to the tangent table to find the tangent value corresponding to degrees or radians.

According to the degree and tangent value of the angle, the tangent curve can be drawn. As shown below.

Combined with the tangent curve, we can summarize several characteristics of the tangent.

1. Periodic – The smallest period is π. tan(θ) = tan(θ + π)
2. Odd function – Since tan(θ) = -tan(θ), the tangent function is an odd function.
3. Monotony – Monotonically increasing in the range.
4. Zero – The tangent at kπ is equal to 0. Here, k is an integer.
5. No maximum and minimum values – It can be clearly seen that the tangent curve extends infinitely upwards or downwards, so the tangent function has no maximum and minimum values. In other words, there are infinite values, which prevents us from recording them.
6. Symmetry – There is no axis of symmetry, only points of symmetry. All (n/2)π  are its centers of symmetry. N is an integer.

Furthermore, in different quadrants of the coordinate axis, the tangent ranges are also different. Tangent comparisons for the four quadrants are listed below.

1. Tangent derivative

The derivative of tangent is equal to the square of secant. Its derivation process is as follows

(tan(θ))’ = (sin(θ)cos(θ))’

= (sin(θ))’cos(θ) – sin(θ)(cos(θ))’cos²(θ)

= cos(θ) * cos(θ) – sin(θ) * (-sin(θ))cos²(θ)

= cos²(θ) + sin²(θ)cos²(θ)

= 1cos²(θ)

=sec²(θ)

2. Inverse tangent

Within a certain range, the inverse tangent function is arctangent, denoted as arctan or tan^-1. It is a type of inverse trigonometric function. The arctangent is used to find the angle value of the known opposite and adjacent sides.

tan(45°) = 1

tan^-1(1) = arctan(1) = 45°

3. Reciprocal tangent

The reciprocal of tangent is cotangent, which is equal to the ratio of its adjacent to its opposite side. The abbreviation is cot.

1tan(θ) = cot (θ)

There is no threshold for using the tangent calculator. You will know it as soon as you learn it, just follow the following two steps.

First, enter a value in degrees or radians.

Second, choose the type, degrees or radians.

Finally, click Calculate button to get the tangent answer, or click Reset button to start a new calculation.

In short, the tangent function plays an important role in trigonometric functions and is often used to solve various triangular problems. It is very necessary to have a tangent calculator, because the tangent calculator can calculate the tangent of any degree or radian, avoiding the shortage of tangent tables.