Graphs of Trigonometric Functions

Overview

In this section, we will explore how to graph the basic trigonometric functions—sine, cosine, and tangent—and understand their properties, such as amplitude, period, phase shift, and vertical shift. These graphs are fundamental to trigonometry and have applications in many real-life scenarios like sound waves, light waves, and mechanical oscillations.

Key Concepts

  • Sine Function: The sine function is periodic with a range of [-1, 1] and a period of \(2\pi\). Its graph oscillates between -1 and 1, crossing the origin at multiples of \( \pi \).
  • Cosine Function: The cosine function is similar to sine, but it starts at its maximum value (1) when \( x = 0 \). Its period is also \( 2\pi \).
  • Tangent Function: The tangent function has a period of \( \pi \) and an undefined value at odd multiples of \( \frac{\pi}{2} \). It has vertical asymptotes at these points.
  • Amplitude: The amplitude is the maximum distance the graph moves from the horizontal axis. For sine and cosine functions, it is the coefficient in front of the function.
  • Period: The period is the length of one complete cycle of the graph. It is determined by the coefficient in front of the variable \( x \). For sine and cosine, the period is \( \frac{2\pi}{\text{coefficient of } x} \).
  • Phase Shift: The phase shift is the horizontal shift of the graph. It is determined by the value inside the function with \( x \), and it shifts the graph left or right.
  • Vertical Shift: The vertical shift moves the graph up or down and is determined by the constant added to the function.

Graphing Basic Trigonometric Functions

Let's examine the graphs of the sine, cosine, and tangent functions:

  • The graph of \( y = \sin(x) \) starts at the origin, oscillates between -1 and 1, and has a period of \( 2\pi \).
  • The graph of \( y = \cos(x) \) starts at 1 and oscillates between -1 and 1, also with a period of \( 2\pi \).
  • The graph of \( y = \tan(x) \) has vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \) (where n is any integer), with a period of \( \pi \).

Transformations of Trigonometric Functions

To transform the graphs of trigonometric functions, we use the following formulas:

  • Amplitude: \( y = A \sin(x) \) or \( y = A \cos(x) \), where \( A \) is the amplitude.
  • Period: \( y = \sin(Bx) \) or \( y = \cos(Bx) \), where the period is \( \frac{2\pi}{|B|} \).
  • Phase Shift: \( y = \sin(x - C) \) or \( y = \cos(x - C) \), where the graph is shifted \( C \) units to the right (if \( C > 0 \)) or left (if \( C < 0 \)).
  • Vertical Shift: \( y = \sin(x) + D \) or \( y = \cos(x) + D \), where \( D \) is the vertical shift.

Practice Problems

  1. Graph the function \( y = 2\sin(x) \) and label its amplitude and period.
    Solution

    Amplitude: 2 (since the coefficient in front of \( \sin(x) \) is 2).

    Period: \( 2\pi \) (since the coefficient of \( x \) is 1, and the period for sine is always \( 2\pi \)).

  2. Graph the function \( y = \cos\left( \frac{x}{2} \right) \) and determine its amplitude and period.
    Solution

    Amplitude: 1 (since there is no coefficient in front of \( \cos(x) \), the default amplitude is 1).

    Period: \( 4\pi \) (since \( B = \frac{1}{2} \), the period is \( \frac{2\pi}{\frac{1}{2}} = 4\pi \)).

  3. Graph the function \( y = \sin(x - \frac{\pi}{4}) \) and identify the phase shift.
    Solution

    Phase shift: \( \frac{\pi}{4} \) units to the right (since the graph shifts to the right by the value of \( C \)).

  4. Graph the function \( y = 3\cos(2x) \) and determine the amplitude, period, and frequency.
    Solution

    Amplitude: 3 (the coefficient of cosine is 3).

    Period: \( \pi \) (since \( B = 2 \), the period is \( \frac{2\pi}{2} = \pi \)).

    Frequency: \( \frac{1}{\pi} \) (frequency is the reciprocal of the period).

  5. Graph the function \( y = -\sin(x) \) and identify the amplitude and phase shift.
    Solution

    Amplitude: 1 (since the amplitude is the absolute value of the coefficient of sine).

    Phase shift: None (there is no horizontal shift, so the phase shift is zero).

  6. Graph the function \( y = \tan(x) \) and describe the locations of the vertical asymptotes.
    Solution

    The vertical asymptotes occur at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer.

  7. Graph the function \( y = \tan(2x) \) and determine its period.
    Solution

    Period: \( \frac{\pi}{2} \) (since \( B = 2 \), the period is \( \frac{\pi}{2} \)).

  8. Graph the function \( y = \cos(x + \frac{\pi}{3}) \) and identify the phase shift.
    Solution

    Phase shift: \( \frac{\pi}{3} \) units to the left (since the graph shifts to the left by the value of \( C \)).

  9. Graph the function \( y = 5\sin(x) \) and label its amplitude and period.
    Solution

    Amplitude: 5 (since the coefficient in front of \( \sin(x) \) is 5).

    Period: \( 2\pi \) (since the coefficient of \( x \) is 1, and the period for sine is always \( 2\pi \)).

  10. Graph the function \( y = \cos(3x) \) and determine its amplitude and period.
    Solution

    Amplitude: 1 (since there is no coefficient in front of \( \cos(x) \), the amplitude is 1).

    Period: \( \frac{2\pi}{3} \) (since \( B = 3 \), the period is \( \frac{2\pi}{3} \)).

→Trigonometric Identities