Solved Examples on Trigonometry Graphs

Example 1: Draw the graph of y = 3 cos 4x + 5.

Solution:

Given: y = 3 cos 4x + 5

Now, compare the given equation with the general form y = a cos (bx + c) + d,

• a = 3, which means the amplitude is 3. (So, the distance between the maximum and minimum value is 6)
• b = 4. Period = 2π/|b| = 2π/|4| = π/2
• c = 0, so there is no phase shift.
• d = 5, which means the graph moved upwards by 5 units.

The graph of y = 3 cos 4x + 5 is given below:

Example 2: Draw the graph of y = cosec x + 3.

Solution:

Given: y = cosec x + 3

• We know that the amplitude of the graph of a cosecant function is undefined as the curve tends to infinity.
• Period = 2π/|b| = 2π/|1| = 2π
• Here, there is no phase shift.
• The graph moved upwards by 3 units.

The graph of y = cosec x + 3 is given below:

Example 3: Draw the graph of y = sin (2x −π) + 2.

Solution:

Given: y = sin (2x − π) + 2

Now, compare the given equation with the general form y = a sin (bx + c) + d,

• a = 1, which means the amplitude is 1. (So, the distance between the maximum and minimum value is 2)
• b = 2. Period = 2π/|2| = 2π/|2| = π
• c = −π. Phase shift = −c/b = − (−π)/2 = π/2
• d = 2, which means the graph moved upwards by 2 units.

The graph of y = sin (2x −π) + 2 is given below:

Example 4: Draw the graph of y = tan x + 1.

Solution:

Given: y = tan x + 1

• We know that the amplitude of the graph of a tangent function is undefined as the curve does not have a maximum or a minimum value and tends to infinity.
• Period = π/|1| = π/|1| = π
• Here, there is no phase shift.
• The graph moved upwards by 1 unit.

The graph of y = tan x + 1 is given below:

Example 5: Draw the graph of y = 2 sin x + 3.

Solution:

Given: y = 2 sin x + 3

Now, compare the given equation with the general form y = a sin (bx + c) + d,

a = 2, which means the amplitude is 2. (So, the distance between the maximum and minimum value is 2)
b = 1. Period = 2π/|1| = 2π/|1| = 2π
c = 0, so there is no phase shift.
d = 3, which means the graph moved upwards by 3 units.

The graph of y = 2 sin x + 3 is given below:

Trigonometric functions study the relationship between the lengths, heights, and angles of right triangles. They are also known as circular functions or angle functions and are widely used in various fields like programming, computing, navigation, solid mechanics, medical imaging, geodesy, measuring the heights of buildings and mountains, etc. As the name implies, trigonometry is referred to as the study of triangles.

There are six trigonometric ratios or functions, which are one of the simplest periodic functions. Sine, cosine, and tangent functions are the most widely used trigonometric functions, whereas their reciprocal functions, cosecant, secant, and cotangent functions, are used less. In this article, we will go through the graphs of the six trigonometric functions in detail.

Let us consider a right-angled triangle ABC, with a right angle at the vertex B, i.e., ∠B = 90°. Let “θ” be the angle at vertex C. Now, the adjacent side/base is the side adjacent to the angle “θ” and the side opposite to the angle “θ” is called the opposite side/perpendicular. The longest side of a right angle, or the side opposite to the right angle, is called a hypotenuse.

• sin θ = Opposite side/Hypotenuse
• cos θ = Adjacent side/Hypotenuse
• tan θ = Opposite side/Adjacent side
• cosec θ = 1/sin θ = Hypotenuse/Opposite side
• sec θ = 1/cos θ = Hypotenuse/Adjacent side
• cot θ = 1/tan θ = Adjacent side/Opposite side