Trigonometric Function Graph

Understanding Trigonometric Function Graphs

Have you ever seen ocean waves? Their up-and-down movement forms patterns that repeat regularly. It turns out that these patterns can be modeled with trigonometric functions.

Before studying trigonometric function graphs, we need to understand angle measurement in radians. In daily life, we are accustomed to using degrees. However, in advanced mathematics, radians are more frequently used.

Converting Degrees and Radians

One complete rotation of a circle is 360° or $2\pi$ radians. This relationship gives us conversion formulas:

Conversion Examples

Converting degrees to radians:

Converting radians to degrees:

What are Amplitude and Period?

Before studying trigonometric function graphs, it's important to understand two key concepts: amplitude and period.

Amplitude

Amplitude is the maximum distance from the center line (x-axis) to the peak or trough of the graph. For functions $y = A \sin x$ or $y = A \cos x$, the amplitude is $|A|$.

Period

Period is the length of interval needed for one complete cycle. For functions $y = \sin(Bx)$ or $y = \cos(Bx)$, the period is $\frac{2\pi}{|B|}$.

General Formulas

For trigonometric functions in the form:

• $y = A \sin(Bx)$ and $y = A \cos(Bx)$:

  $$y = A \sin(Bx)$$

  $$y = A \cos(Bx)$$

• $y = A \tan(Bx)$:

  $$y = A \tan(Bx)$$

  $$\text{Amplitude} = \text{undefined (infinite)}$$

  $$\text{Period} = \frac{\pi}{|B|}$$

Sine Function Graph

The function $y = \sin x$ is a periodic function with period $2\pi$. This means its graph pattern repeats every $2\pi$ interval.

Characteristics of $y = \sin x$ graph:

• Period: $2\pi$ (graph repeats every $2\pi$ units)
• Amplitude: 1 (maximum - minimum value divided by 2)
• Domain: All real numbers
• Range: $[-1, 1]$
• x-intercepts: $x = n\pi$ where $n$ is an integer
• Maximum value: 1 at $x = \frac{\pi}{2} + 2n\pi$
• Minimum value: -1 at $x = \frac{3\pi}{2} + 2n\pi$

Cosine Function Graph

The function $y = \cos x$ has a shape similar to sine, but shifted $\frac{\pi}{2}$ to the left.

Characteristics of $y = \cos x$ graph:

• Period: $2\pi$
• Amplitude: 1
• Domain: All real numbers
• Range: $[-1, 1]$
• x-intercepts: $x = \frac{\pi}{2} + n\pi$
• Maximum value: 1 at $x = 2n\pi$
• Minimum value: -1 at $x = \pi + 2n\pi$

Comparison of Sin and Cos

Tangent Function Graph

The function $y = \tan x$ differs from sin and cos because it has vertical asymptotes.

Characteristics of $y = \tan x$ graph:

• Period: $\pi$ (shorter than sin and cos)
• Amplitude: Undefined
• Domain: $x \neq \frac{\pi}{2} + n\pi$
• Range: All real numbers
• Vertical asymptotes: $x = \frac{\pi}{2} + n\pi$
• x-intercepts: $x = n\pi$

Transformations of Trigonometric Functions

Amplitude Changes

The function $y = A \sin x$ changes the amplitude to $|A|$.

Period Changes

The function $y = \sin(Bx)$ changes the period to $\frac{2\pi}{|B|}$.

Vertical and Horizontal Shifts

General form:

• $A$: Amplitude
• $B$: Affects period ( $\text{period} = \frac{2\pi}{|B|}$)
• $C$: Horizontal shift (phase)
• $D$: Vertical shift

Notice the horizontal and vertical shifts of the graph:

Exercises

1. Convert the following angles:

   • 120° to radians
   • $\frac{5\pi}{6}$ radians to degrees

2. Determine the period and amplitude of:

   • $y = 3 \sin(2x)$
   • $y = -2 \cos(\frac{x}{3})$

3. Sketch the graph of $y = 2 \sin(x + \frac{\pi}{3}) - 1$. Determine:

   • Amplitude
   • Period
   • Phase shift
   • Vertical shift

4. If tidal height is modeled by $h(t) = 2 \sin(\frac{\pi t}{6}) + 5$ meters, where t is in hours:

   • What are the maximum and minimum water heights?
   • What is the tidal period?

5. Determine the equation of a trigonometric function that has:

   • Amplitude 3
   • Period $\pi$
   • Shifted $\frac{\pi}{4}$ to the right
   • Shifted 2 units up

Answer Key

1. Angle conversion:

   • $120° = 120 \times \frac{\pi}{180} = \frac{2\pi}{3}$ radians
   • $\frac{5\pi}{6} = \frac{5\pi}{6} \times \frac{180}{\pi} = 150°$

2. Period and amplitude:

   • $y = 3 \sin(2x)$: Amplitude = 3, Period = $\frac{2\pi}{2} = \pi$
   • $y = -2 \cos(\frac{x}{3})$: Amplitude = 2, Period = $\frac{2\pi}{1/3} = 6\pi$

3. For $y = 2 \sin(x + \frac{\pi}{3}) - 1$:

   • Amplitude: 2
   • Period: $2\pi$
   • Phase shift: $\frac{\pi}{3}$ to the left
   • Vertical shift: 1 unit down

4. For $h(t) = 2 \sin(\frac{\pi t}{6}) + 5$:

   • Maximum height: 5 + 2 = 7 meters
   • Minimum height: 5 - 2 = 3 meters
   • Period: $\frac{2\pi}{\pi/6} = 12$ hours

5. Equation that satisfies the requirements:

   $$y = 3 \sin(2(x - \frac{\pi}{4})) + 2$$

   or

   $$y = 3 \sin(2x - \frac{\pi}{2}) + 2$$