Graphs of Trigonometric Functions

Graphing trigonometric functions means plotting sine, cosine, tangent, and their inverse functions to observe their behavior across various angles. The graph also illustrates properties such as periodicity, amplitude, and phase shifts in them.

In these graphs:

• The x-axis represents the angles, typically measured in radians (0,  ​${\dfrac{\pi }{2}}$, ${\pi}$, ${\dfrac{3\pi }{2}}$, 2π)
• The y-axis represents f(x), the value of the trigonometric function corresponding to each angle.

The graphs are periodic, which means they repeat their patterns after a fixed interval called the period.

Sine (y = sin x)

The graph of the sine function is closely related to the unit circle, a circle of radius 1, and centered at the origin in the xy-plane. Every point on this circle is given by the coordinates (x, y) = (cos θ, sin θ), satisfying the equation:

x² + y² = 1

To construct the graph of y = sin⁡ x, we use the y-coordinate of points on the unit circle, which corresponds to the sine of the angle θ. Thus, the sine function can be expressed as:

f(θ) = sin θ or y = sin x 

Here, we consider the angles θ = 0, ${\dfrac{\pi }{4}}$, ​${\dfrac{\pi }{2}}$, ${\dfrac{3\pi }{4}}$, ${\pi}$. The horizontal lines from the unit circle to the sine graph show how these values are mapped.

Extending this process to angles from 0 to 2π radians, we can trace the sine graph for a full cycle. The resulting graph oscillates between −1 and 1, capturing the periodic nature of the sine function.

Since trigonometric functions are periodic, they repeat its pattern after every 2π radians (or 360°). This means the graph of y = sin⁡x can be extended indefinitely, showing multiple cycles of its wave-like structure. The key features of its graph are listed below:

Domain: (-∞, ∞)

Range: [-1, +1]

Amplitude: 1

Period: 2π

X-Intercept: x = nπ, for all n

Y-Intercept: y = 0

Line of symmetry: At the origin (0, 0)

Type of function: Odd function

Cosine (y = cos x)

The graph of the cosine function, like the sine function, is also related to the unit circle. On this circle, each point is represented by the coordinates (x, y) = (cos ⁡θ, sin ⁡θ ), where the x-coordinate corresponds to the cosine of the angle θ. This relationship is derived from the equation:

x² + y² = 1

Thus, the cosine function can be defined as:

f(θ) = cos θ or y = cos x 

Since the cosine function is phase-shifted compared to the sine function:  

cos x = sin(x + 90°) for all x. This implies:

• cos 0° has the same value as sin 90°,
• cos 90° has the same value as sin180°,
• cos 180° has the same value as sin 270°, and so on.

Thus, the graph of the cosine function is identical to the graph of the sine function but shifted to the left by 90° (​or ${\dfrac{\pi }{2}}$ radians), as shown.

The graph of the cosine function has the following properties: 

Domain: (-∞, ∞)

Range: [-1, +1]

Amplitude: 1

Period: 2π

X-Intercept: x = ${\left( 2n+1\right) \dfrac{\pi }{2}}$, for all n

Y-Intercept: y = 1

Line of symmetry: The y-axis

Type of function: Even function

Tangent (y = tan x)

The tangent function is defined as: 

tan x = ${\dfrac{\sin x}{\cos x}}$

By applying this definition, it is graphed as follows:

A tangent function has the following properties:

Domain: ℝ – ${\left( 2n+1\right) \dfrac{\pi }{2}}$

Range: (-∞, ∞)

Period: π

X-Intercept: x = nπ, for all n

Y-Intercept: y = 0

Line of symmetry: At the origin (0, 0)

Vertical asymptotes: x = ${\left( 2n+1\right) \dfrac{\pi }{2}}$

Type of function: Odd function

Cosecant (y = cosec x)

The cosecant function, y = cosec x, is the reciprocal of the sine function. It is undefined wherever sin x = 0, creating vertical asymptotes at these points. Between the asymptotes, its graph forms distinct parabolic curves that extend upward and downward from the peaks (highest points) and troughs (lowest points) of the sine graph, as shown.

The cosecant function has the given properties:

Domain: ℝ – nπ

Range: (-∞, -1] ∪ [1, ∞)

Period: 2π

X-Intercept: It does not exist

Y-Intercept: It does not exist

Line of symmetry: At the origin (0, 0)

Vertical asymptotes: x = nπ

Type of function: Odd function

Secant (y = sec x)

The secant function, y = sec x, is the reciprocal of the cosine function. Like cosecant, it has vertical asymptotes wherever cos x = 0. The graph consists of parabolic curves, which are centered around the peaks and troughs of the cosine graphs, as shown.

The secant function has

Domain: ℝ – ${\left( 2n+1\right) \dfrac{\pi }{2}}$

Range: (-∞, -1] ∪ [1, ∞)

Period: 2π

X-Intercept: It does not exist

Y-Intercept: y = 1

Line of symmetry: The y-axis

Vertical asymptotes: x = ${\left( 2n+1\right) \dfrac{\pi }{2}}$

Type of function: Even function

Cotangent (y = cot x)

The cotangent function has a period of π and is undefined wherever tan x = 0, creating vertical asymptotes at these points. Unlike tangent, the graph of cotangent slopes downward between the asymptotes, which makes a repeating pattern with a distinct downward tilt.

The tangent function follows the given properties:

Domain: ℝ – nπ

Range: (-∞, ∞)

Period: π

X-Intercept: x = ${\left( 2n+1\right) \dfrac{\pi }{2}}$, for all n

Y-Intercept: It does not exist

Line of symmetry: At the origin (0, 0)

Vertical asymptotes: x = nπ

Type of function: Odd function

Transformations

Trigonometric graphs can be adjusted in various ways to modify their shape and position by changing their amplitude, period, phase, and vertical shift. They are used to show periodic patterns, analyze the behavior of functions, and for solving complex problems.

Amplitude

The amplitude determines the vertical stretch or compression of the graph. It represents the height of the peaks and the depth of the troughs from the midline. 

Mathematically, the amplitude is the absolute value of the number multiplied by the trigonometric function. 

In a function y = A sin x, 

• |A| is the amplitude of the sine graph
• Larger values of A increase the height of peaks and troughs, while smaller values compress them.

For example, if y = 2 sin x, then the function has peaks at 2 and troughs at -2

Note: The orientation of the graph depends on the sign of the amplitude: 

• A positive amplitude keeps the graph in its usual orientation.
• A negative amplitude flips the graph upside down.

Period

The period defines the horizontal length of one complete cycle of a trigonometric function. Mathematically, it is calculated using the formula:

${\omega =\dfrac{\langle regular \  period\rangle }{\left| B\right| }}$

Here, B is the angular coefficient.

• Since sines and cosines (and their reciprocals, cosecants, and secants) have a regular period of 2π. Thus, their period formula is: ${\omega =\dfrac{2\pi }{\left| B\right| }}$
• Since tangent and its reciprocal cotangent have a regular period of π. Thus, its period formula is: ${\omega =\dfrac{\pi }{\left| B\right| }}$

For example, if y = sin 2x has a period of ${\omega =\dfrac{2\pi }{2}}$ = π. This means the graph of y = sin 2x completes two cycles within an interval of [0, 2π]

Phase Shift

Phase shift refers to the horizontal movement of a graph from its usual position. 

In a function y = sin(x – C), the phase shift is determined by C. The direction of the shift is related to the sign of C: 

• A positive C shifts the graph to the right
• A negative C shifts the graph to the left

For example, y = sin(x – π), the phase shift moves the graph to the right by π units.

Vertical Shift

A vertical shift moves the graph up or down along the y-axis, changing its midline. 

In a function y = sin x + D, the vertical shift is determined by D, and its direction depends on the sign of D:

• A positive D shifts the graph upward
• A negative D shifts the graph downward

For example, y = sin x + 2 raises the midline to y = 2.

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