How to Find Phase Shift of Tangent Function

Tangent functions are trigonometric functions that are used to represent periodic oscillatory phenomena such as sound waves, light waves, and sine waves. These functions have a phase shift that determines how the wave is shifted in space. The phase shift refers to the amount by which the wave is shifted to the right or left of the origin. In this article, we will discuss how to find phase shift of tangent function.

What is a Phase Shift?

A phase shift is an angle that determines how much a wave is shifted to the right or left of the origin. It is measured in radians and is denoted by the variable ‘ϕ’. The phase shift of a tangent function determines the horizontal shift of the function.

How to Find the Phase Shift of a Tangent Function?

The general equation of a tangent function is given by:

y = A tan(Bx – C) + D

Here,

A = amplitude
B = period
C = phase shift (horizontal shift)
D = vertical shift

To find the phase shift of a tangent function, follow these steps:

Step 1: Identify the value of ‘C’ in the equation.

Step 2: Convert the value of ‘C’ into radians if it is given in degrees. To convert degrees into radians, use the formula:

Radians = Degrees x π/180

Step 3: Divide the value of ‘C’ by ‘B’ to get the horizontal shift. The horizontal shift is given by:

Horizontal shift = C/B

Step 4: The phase shift is the opposite of the horizontal shift. Therefore, to get the phase shift, multiply the horizontal shift by ‘-1’. The phase shift is given by:

Phase shift = – (C/B)

Let’s take an example to illustrate the process of finding the phase shift of a tangent function.

Example:

Find the phase shift of the tangent function:

y = 2 tan(3x – 45°) + 1

Solution:

Step 1: Identify the value of ‘C’ in the equation.

Here, C = 45°

Step 2: Convert the value of ‘C’ into radians.

Radians = 45° x π/180
Radians = 0.7854

Step 3: Divide the value of ‘C’ by ‘B’.

Here, B = 3

Horizontal shift = C/B
Horizontal shift = 0.7854/3
Horizontal shift = 0.2618

Step 4: The phase shift is the opposite of the horizontal shift. Therefore,

Phase shift = – (C/B)
Phase shift = – (0.7854/3)
Phase shift = -0.2618

Therefore, the phase shift of the given tangent function is -0.2618 radians.

FAQs

Q.1. What is the difference between phase shift and vertical shift?

A. The phase shift determines how much a wave is shifted to the right or left of the origin, whereas the vertical shift determines how much a wave is shifted up or down from the origin.

Q.2. What are the properties of tangent functions?

A. The properties of tangent functions are:

– Tan (x) is odd; that is, tan (-x) = – tan (x).
– The tangent function is periodic with a period of π.
– The tangent function is undefined at odd multiples of π/2.