Tangent

Tangent, often denoted as "tan," is one of the three main trigonometric functions, with the other two being sine (sin) and cosine (cos). The trigonometric functions can be defined in some different ways. Most commonly, they are defined based on the ratio of the sides of a right triangle, or based on a unit circle.

In terms of a right triangle, the tangent of an angle θ, denoted tan(θ), is the ratio of the length of the opposite edge to the length of the adjacent edge of a right triangle. Referencing the triangle above, the side opposite of θ is side a and the side next to θ is side b. Thus:

Unit-circle definition

The tangent function can also be defined using a unit circle (a circle with radius of 1 unit). In a plane with a unit circle centered at the origin of a coordinate system, a ray from the origin forms an angle θ with the x-axis. The tangent value of θ is represented in the figure below by the blue line. It is the length of the line segment formed by the intersections of line x = 1 with x-axis and x = 1 with the ray. Unlike the definitions of the trigonometric functions based on right triangles, this definition works for any angle, not just the acute angles of right triangles.

Example:

Refer to the triangle at the top of the page. Jack is standing 17 meters from the base of a tree. The distance between Jack and the tree is the length b in our reference triangle. Given that the angle from Jack’s feet to the top of the tree, θ in the reference triangle, is 51°, what is the height of the tree? Assuming that the tree falls towards Jack exactly along the line b, will the tree fall on Jack?

Below are our known values:

θ = 51°
b = 17
a = ?

This example can be solved using tangent as follows:

a = 17 × tan(51°)

Using a calculator, tan(51°) = 1.235, so:

a = 17 × 1.235 = 20.995

Therefore, the height of the tree is 20.995 m, and given that Jack is rooted by fear, the tree will fall on him.

Tangent function properties

Like cosine and sine, the tangent function is a periodic function, and thus has certain properties based on its periodicity. Below are some of the more commonly used properties:

tan(-θ) = -tan(θ)
tan(180° - θ) = -tan(θ)
tan(θ + n×180°) = tan(θ)
tan(90° - θ) = cot(θ)
cot(90° - θ) = tan(θ)
tan(90° + θ) = -cot(θ)
cot(90° + θ) = -tan(θ)

Example:

Given tan(40°) = 0.839, what is tan(140°)?

tan(140°)= tan(180° - 40°)
 = -tan(40°) = -0.839

Example:

Given tan(55°) = 1.428, what is tan(595°)?

tan(595°)= tan(55° + 3×180°)
 = tan(55°) = 1.428

Values of the tangent function

The value of the tangent is related to that of sine and cosine in that . Tangent is positive when both sine and cosine are positive, or both sine and cosine are negative. This happens between 0° and 90° and 180° and 270°. Tangent is negative between 90° and 180° and 270° and 360° because either sine or cosine is negative in those angle ranges. Tangent has a period of 180° and asymptotes at odd multiples of 90°. In other words, when referencing the unit circle, tangent is positive in quadrants I and III and negative in quadrants II and IV.

Below is a table of tangent values for commonly used angles:

Angle in degreesAngle in radiansTangent valueTangent value in decimals
000
15°0.268
30°0.577
45°11
60°1.732
75°3.732
90°UndefinedUndefined
180°π00
270°UndefinedUndefined
360°2π00

Tangent calculator:

The following is a calculator to find out either the tangent value of an angle or the angle from the tangent value.

tan =

Law of tangents

The law of tangents is an equation relating the tangents of two angles of a triangle and the length of the opposing sides of those angles. Referencing the triangle below, the law of tangents is written as:

In the equation, A and B are the angles opposite of sides of length a and b. The law of tangents can be used in any case where two angles and a side or two sides and an angle are known to determine the fourth value.

Example:

Given side a = 7, angle A = 30°, and angle B = 65°, use the law of tangents to find all the sides and angles of the triangle.

Since the angles of a triangle add to 180°, angle C = 180° - 30° - 65° = 85°.

Using the law of tangents,

7 - b = -0.289(7 + b)
7 = -2.023 + 0.711b

We can use the same method to find side c, using either a or b as the second side and respective angle to find that:

c = 13.941

Inverse function of tangent

While the tangent function is used to determine a ratio from a given angle, its inverse, written arctan or tan-1, is used to compute an angle from a given ratio:

Example:

Using the right triangle above and given that a = 3 and b = 5, what is the angle θ?

We can determine the angle θ as follows:


See also sine, cosine, unit circle, trigonometric functions, trigonometry.