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Graphs of trigonometric functions

Mathematics - Grade 11 · Trigonometry

Graphs of Trigonometric Functions

Trigonometric functions are periodic functions. This means they repeat their values in regular intervals. The three main trigonometric functions are sine (sin), cosine (cos), and tangent (tan). Each of these functions has a unique graph with specific characteristics.

Sine Function

The sine function is defined as the ratio of the opposite side to the hypotenuse in a right triangle. The graph of the sine function is a smooth wave that oscillates between -1 and 1.

xyO−300−200−100100200300−1.5−1−0.50.51B(0, 0)y = sin(x)
The graph of f(x) = sin(x)

Key Features of the Sine Graph

  • Amplitude: The maximum height of the wave. For sin(x), the amplitude is 1.
  • Period: The length of one complete cycle of the wave. The period of sin(x) is 360° or 2π radians.
  • Intercepts: The graph crosses the x-axis at 0°, 180°, and 360°.
  • Maximum and Minimum Points: The maximum point is at 90° (1) and the minimum point is at 270° (-1).

Cosine Function

The cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. The graph of the cosine function is also a smooth wave, similar to the sine graph, but it starts at its maximum value.

xyO−300−200−100100200300−1.5−1−0.50.51E(0, 1)y = cos(x)
The graph of f(x) = cos(x)

Key Features of the Cosine Graph

  • Amplitude: The amplitude is also 1 for cos(x).
  • Period: The period is 360° or 2π radians.
  • Intercepts: The graph crosses the x-axis at -90°, 90°, and 270°.
  • Maximum and Minimum Points: The maximum point is at 0° (1) and the minimum point is at -180° (-1).

Tangent Function

The tangent function is defined as the ratio of the sine to the cosine function. The graph of the tangent function has a different shape and is not continuous; it has vertical asymptotes.

xyO−300−200−100100200300−10−8−6−4−22468C(0, 0)y = tan(x)
The graph of f(x) = tan(x)

Key Features of the Tangent Graph

  • Amplitude: The tangent function does not have an amplitude.
  • Period: The period of tan(x) is 180° or π radians.
  • Intercepts: The graph crosses the x-axis at multiples of 180° (0°, 180°, etc.).
  • Vertical Asymptotes: The graph has vertical asymptotes at odd multiples of 90° (-90°, 90°, -270°, 270°).

Transformations of Trigonometric Functions

Trigonometric functions can be transformed by changing their amplitude, period, phase shift, or vertical shift. The general form of a transformed trigonometric function is:

y = a * sin(b(x - c)) + d
  • a: Amplitude (vertical stretch or compression).
  • b: Affects the period. The period is calculated as 360°/b.
  • c: Phase shift (horizontal shift). Positive values shift the graph to the right.
  • d: Vertical shift (moves the graph up or down).

Example of Transformation

Consider the function:

y = 2 * sin(2(x - 30)) + 1

In this case:

  • The amplitude is 2.
  • The period is 360°/2 = 180°.
  • The phase shift is 30° to the right.
  • The vertical shift is 1 unit up.

Summary

  • The sine function oscillates between -1 and 1 with a period of 360°.
  • The cosine function oscillates between -1 and 1 with a period of 360°.
  • The tangent function has a period of 180° and includes vertical asymptotes.
  • Transformations can change the amplitude, period, phase shift, and vertical shift of the trigonometric functions.

Check your understanding

  1. What is the period of the sine function?
  2. Identify the maximum and minimum points of the cosine function.
  3. Explain how to find the vertical asymptotes of the tangent function.
  4. What effect does the parameter 'a' have in the transformation of a sine function?