Tangent Function

Tangent Function: Trigonometric function tan(x) is called a tangent function it is one of the main six trigonometric functions and is generally written as tan x. The tangent function is the ratio of the opposite side and the adjacent side of the angle in consideration in a right-angled triangle.

In this article, we will learn about Tangent function definition, Tangent function graph, tangent function values, examples, and others in detail.

What is Tangent Function?

Tangent function, also called the tan(x), is a trigonometric function that takes the ratio of the side opposite to the angle being considered in a right-angled triangle to its adjacent side. We have six trigonometric functions and tan(x) is one of them. Various trigonometric formulas and identities are used in solving trigonometric problems.

Tangent function is a periodic function and the period of y = a tan(bx) is given as,

Period = π/|b|

Tangent Function Formula

Tangent function (tan) is the ratio of the sine (sin) and cosine (cos) functions, which are commonly used in trigonometry. Mathematically, it can be expressed as:

tan(θ) = sin(θ)/cos(θ)

Alternatively, if you know the lengths of the sides of a right triangle (opposite and adjacent), you can use the formula:

tan(θ) = Opposite Side/Adjacent Side = Perpendicular/Base

👁 Right Angled Triangle

A right-angle triangle with angle of consideration as θ is shown in the image added above.

Also Read: Tangent Formula

Tangent Function Graph

Graph for y = tan (x) = y shows how the tangent returns a value y for the angle x (measured in radians). Tangent function is a periodic function and the period of tangent function is π radians, thus the graph of tangent function repeat itself in every π radians along the x-axis. The graph for the tangent function is added below:

👁 Graph of Tangent Function

Tangent Function Values

Values of the tangent function for some common angles can be learnt using the table added below:

Degrees	Radians	sin(θ)	cos(θ)	tan(θ)
0°	0	0	1	0
30°	π/6	1/2	√3/2	√3/3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	Undefined

The tan function is undefine at 90 degrees because division by zero is not possible. The adjacent side of a right triangle would be zero.

Tangent Function Identities

Tan (tangent) function has various identities which are used in solving various trigonometric problems Some of the important identities of tangent function are:

1. Reciprocal Identity

Tangent function is the reciprocal of cotangent function, i.e.

tan(θ) = 1/cot(θ)

2. Quotient Identity

Tangent function is the ratio of sine function and cosine function, i.e.

tan(θ) = sin(θ)/cos(θ)

3. Pythagorean Identity

Pythagorean identity for trigonometric tan function is:

tan²(θ)+1 = sec²(θ)

4. Addition and Subtraction Formulas

Addition and subtraction formulas for tangent function is:

tan(A+B) = tan(A) + tan(B)/1 - tan(A)tan(B)

tan(A-B) = tan(A) - tan(B)/1 + tan(A)tan(B)

5. Double Angle Identity

Double angle identity for tangent function is:

tan(2θ) = 2tan(θ)/1 - tan²(θ)

6. Half-Angle Identity

Half angle identity for tangent function is:

tan(θ/2) = 1 - cos(θ)/sin(θ)

Tangent Function Domain and Range

Domain of Tangent Function:

Domain of tangent function consists of all real numbers except at points where cossine function is zero as the tangent function is the ratio of sine and cosine function and division by zero is undefined. Therefore, the domain of tangent function is given by:

Domain of tan(θ) = R - {(2k+1) π/2}, where k is an integer.

Range of Tangent Function:

Range f tangent function is the real line. So, the range of the tangent function is:

Range of tan(θ) = R, where R is set of real numbers.

In summary, the domain of the tangent function excludes odd multiples of π/2, where it is undefined, while its range includes all real numbers.

Tangent Function Period

​The tangent function, denoted as tan (x) is a periodic function with a period of π radians or 180∘. This means that the function repeats its values every π radians along the x-axis. Mathematically, this property can be expressed as:

tan(x+π)=tan(x)

Properties of Tangent Function

Some properties of the tangent function includes:

• Periodicity: Tangent function is periodic with a period of π (which is the same for 180 degrees).
• Odd Function: Since tangent function satisfies the property tan(-θ) = -tan(θ), hence it is an odd function.
• Vertical Asymptotes: Tangent function is defined as a vertical asymptote at any odd multiple of π/2 radians (90 degrees).
• Domain: Domain of tangent function is all real number expect odd multiples of π/2.
• Range: Range of tangent function is the all the real number.
• Relationship with Sine and Cosine: Tangent function is the ratio of sine function and cosine function, i.e. tan(θ) = sin(θ)/cos(θ).
• Symmetry: Tangent function is symmetric along the origin.
• Slope of a straight line is the tangent of the angle made by the line with the positive x-axis.

Check: Domain and Range of Trigonometric Functions | Trig & Inverse Trig

Inverse Tangent Function

Inverse trigonometric function is called arctan. Mathematically is represented as, “tan^-1 (x)” or “arctan x”.

Also,

• Domain of tan inverse x is (−∞,∞)
• Range of taninverse x is [-π/2, π/2]

People Also Read:
Sine Function	Cosine Function
Secant Function	Cosecant Function
Tangent Formulas	Tan Theta Formula

Tangent Function Examples

Example 1: Let sec x = 5/3. If the required angle x is located in the I quadrant, then find the value of tan x.

Solution:

sec(x) = 1/cos(x), therefore

• sec(x) = 5/3
• cos(x) = 3/5

Using Pythagorean Identity for cosine and sine:

sin²x + cos²x = 1

We can write:

sin²(x) = 1- cos²(x)

sin²(x) = 1- (3 / 5)²

sin²(x) = 1 - (9 / 25)

sin²(x) = 16 / 25

sin(x) = 4/5 (for I quadrant)

Using definition of tangent function,

tan (x) = sin (θ)/cos (θ)

tan(x) = (4/5)/(3/5) = 4/3

So, tan(x) = 4/3

Example 2: Simplify tan^-1[2cos {sin^–1(1/2}].

Solution:

Given,

= tan^-1[2cos {sin^–1(√3/2)}]

= tan^-1 [2cos (sin^–1(sinπ/3))]

= tan^-1 [2cos (π/3)]

= tan^-1 [2×1/2]

= tan^-1(1)

= tan^-1{tan (π/4)

= (π/4)

tan^-1[2cos {sin^–1(1/2}] = π/4

Example 3: Find the length of the shadow formed by a tree with 15 ft height on an horizontal plane, when the elevation of the sun from the horizon is exactly 90°.

Solution:

tangent of 60 degrees is (height of tree)/(length of shadow)

Given,

• Height of Tree = 15 ft

tan(60^∘) = 15/x

{tan(60^∘) = √3}

√3 = 15/x

x = 15/√3

x = 5√3 ft

Hence, length of the shadow is 1.74 m ( ft)

Example 4: Find base of right angle triangle if its perpendicular is 4 cm and angle of consideration is 45°.

Solution:

Given,

• Perpendicular = 4 cm
• θ = 45°

tan 45° = Perpendicular/Base

1 = 4/Base

Base = 4 cm

Thus, base of right angle triangle is 4 cm.

Practice Problems on Tangent Function

Q1: Given tan(x) = 4/5, find the value of x in degrees.

Q2: Find the solution of tan(x) = −1 for the closed interval [0, 2π] in the equation tan(x) = −1.

Q3: If tan(x) = √3, where we have to solve it in degrees, find the general solution for x.

Q4: Therefore, cot(90 - A) = 3/2, the question is: find the value of tan A.