Graphs of Trigonometric Functions (AQA A-Level Mathematics): Revision Notes

5.2.1 Graphs of Trigonometric Functions

Graphs of Trigonometric Functions are fundamental in understanding the behaviour of sine, cosine, and tangent functions. These graphs illustrate the periodic nature of these functions and are essential in both pure and applied mathematics.

1. Graph of the Sine Function: $\ y = \sin x$

• Shape: The graph of $\ y = \sin x$ is a smooth, continuous wave that oscillates above and below the $x$-axis.
• Period: The sine function has a period of $\ 360^\circ \ (or \ 2\pi \ radians),$ meaning the pattern repeats every $\ 360^\circ$.
• Amplitude: The amplitude is the maximum distance from the $x$-axis, which is $1$ for the basic sine function. The graph oscillates between $\ y = 1 \ and \ y = -1 .$

Graph Characteristics:

• $X$-Intercepts: $\ x = 0^\circ, 180^\circ, 360^\circ, \dots$
• Maximum Points:$\ x = 90^\circ, 450^\circ, \dots \ where \ y = 1$
• Minimum Points: $\ x = 270^\circ, 630^\circ, \dots \ where \ y = -1$

2. Graph of the Cosine Function: $\ y = \cos x$

• Shape: The graph of $\ y = \cos x$ is also a smooth, continuous wave, similar to the sine graph but shifted horizontally.
• Period: Like sine, the cosine function has a period of $\ 360^\circ \ (or \ 2\pi \ radians).$
• Amplitude: The amplitude is $1$, so the graph oscillates between $\ y = 1 \ and \ y = -1 .$

Graph Characteristics:

• X-Intercepts: $\ x = 90^\circ, 270^\circ, 450^\circ, \dots$
• Maximum Points: $\ x = 0^\circ, 360^\circ, \dots \ where \ y = 1$
• Minimum Points:$\ x = 180^\circ, 540^\circ, \dots \ where \ y = -1$

3. Graph of the Tangent Function: $\ y = \tan x$

• Shape: The graph of $\ y = \tan x$ is quite different from sine and cosine. It has a repeating pattern but includes vertical asymptotes where the function is undefined.
• Period: The tangent function has a period of $\ 180^\circ \ (or \ \pi \ radians)$, so it repeats every $\ 180^\circ .$

Graph Characteristics:

• X-Intercepts: $\ x = 0^\circ, 180^\circ, 360^\circ, \dots$
• Asymptotes: $\ x = 90^\circ, 270^\circ, \dots$
• The graph increases from $\ -\infty \ to \ \infty$ between each pair of asymptotes.

4. Transformations of Trigonometric Graphs

Trigonometric graphs can be transformed in several ways:

• Vertical Stretching/Compressing:
  • $\ y = A \sin x \ or \ y = A \cos x ,$ where $\ A$ affects the amplitude.
  • Example: $\ y = 2 \sin x$ doubles the amplitude.
• Horizontal Stretching/Compressing:
  • $\ y = \sin(Bx) \ or \ y = \cos(Bx) \, where \ B$ affects the period.
  • The new period becomes $\ \frac{360^\circ}{B} \ (or \ \frac{2\pi}{B} \ radians).$
  • Example:$\ y = \sin(2x)$ compresses the graph, halving the period.
• Vertical Shifting:
  • $\ y = \sin x + C \ or \ y = \cos x + C \, where \ C$shifts the graph up or down.
  • Example:$\ y = \sin x + 1$ shifts the graph $1$ unit up.
• Horizontal Shifting:
  • $\ y = \sin(x - D) \ or \ y = \cos(x - D) \, where \ D$ shifts the graph left or right.
  • Example: $\ y = \sin(x - 30^\circ)$ shifts the graph $30°$ to the right.

Summary:

Understanding the basic shapes and characteristics of the sine, cosine, and tangent graphs, as well as how they transform, is crucial for solving trigonometric problems. These functions are periodic and can model a wide range of real-world phenomena, from sound waves to seasonal patterns.