Trigonometric graphs show the repeating behavior of functions like sine, cosine, and tangent. Each graph has important features such as period, amplitude, symmetry, and intercepts, which help describe its shape and behavior.
The period of the sine function is 2π and its zeros are at every integer multiple of π. It is symmetric about the origin, making it an odd function.
θ
0°
90°
180°
270°
y = sin θ
0
1
0
−1
Domain
(-∞, + ∞)
Range
[-1, +1]
Amplitude
1
X−intercept
x=nπ, ∀n
Y−intercept
y = 0
Cosine Function
The period of the cosine function is 2π and its zeros fall at odd multiples of π/2. It is the sine graph shifted π/2 units to the left, and is symmetric about the Y-axis, making it an even function.
θ
0°
90°
180°
270°
y = cos θ
1
0
−1
0
Domain
(-∞, + ∞)
Range
[-1, +1]
Amplitude
1
X−intercept
x = (2n + 1)π/2, ∀n
Y−intercept
y = 1
Tangent Function
The tangent function is undefined at odd multiples of π/2 where cos θ = 0, creating vertical asymptotes there. Its period is π and its range is all real numbers since it has no maximum or minimum.
θ
-45°
0°
45°
90°
y = tan θ
-1
0
1
undefined
Domain
R - (2n + 1)π/2
Range
(-∞, +∞)
X−intercept
x=nπ, ∀n
Y−intercept
y=0
Line of symmetry
Origin
Vertical asymptotes
x = (2n + 1)π/2
Type of function
Odd function
Cotangent Function
The cotangent function is the reciprocal of tangent, undefined at every integer multiple of π where sin θ = 0. Its period is π and it has no Y-intercept since x = 0 is one of its undefined points.
θ
0°
45°
90°
135°
y = cot θ
undefined
1
0
-1
Domain
R - nπ
Range
(-∞, +∞)
X−intercept
x = (2n + 1)π/2, ∀n
Line of symmetry
Origin
Vertical asymptotes
x = nπ
Type of function
Odd function
Cosecant Function
The cosecant function is the reciprocal of sine, undefined at every integer multiple of π. Its period is 2π and it has no intercepts of any kind. Its range excludes (−1, 1), taking only values of magnitude 1 or greater.
θ
0°
90°
180°
270°
y = csc θ
undefined
1
undefined
−1
Domain
R - nπ
Range
(-∞, -1] U [+1, +∞)
Line of symmetry
Origin
Vertical asymptotes
x = nπ
Type of function
Odd function
Secant Function
The secant function is the reciprocal of cosine, undefined at odd multiples of π/2. Its period is 2π and its range also excludes (−1, 1). It inherits cosine's symmetry about the Y-axis, making it an even function.
θ
0°
90°
180°
270°
y = sec θ
1
undefined
−1
undefined
Domain
R - (2n + 1)π/2
Range
(-∞, -1] U [+1, +∞)
Y−intercept
y = 1
Line of symmetry
Y-axis
Vertical asymptotes
x = (2n + 1)π/2
Important Features
For every trigonometric graph, there are important features:
Amplitude: Amplitude is half of the distance between the maximum value and the minimum value, or the height of the curve from the center line.
Vertical Shift: The displacement of the graph up or down (parallel to the y-axis).
Period: The period is the distance between the repetitions of any function.
Phase: The position of the waveform at a fraction of a period is referred to as its phase, and it is expressed in angles or radians.
Phase Shift: The displacement of the graph parallel to the x-axis (horizontal shift).
Solved Examples
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.
