Question

Without graphing, determine the amplitude and period of each function. State the period in degrees and in radians. a) \(y=2 \sin x\) b) \(y=-4 \cos 2 x\) c) \(y=\frac{5}{3} \sin \left(-\frac{2}{3} x\right)\) d) \(y=3 \cos \frac{1}{2} x\)

Short answer

a) Amplitude: 2, Period: 360° or 2\pi radians. b) Amplitude: 4, Period: 180° or \pi radians. c) Amplitude: \(\frac{5}{3}\), Period: 540° or 3\pi radians. d) Amplitude: 3, Period: 720° or 4\pi radians.

Step-by-step solution

Identify amplitude and period for y=2 \sin x

For the function \(y = A \sin(Bx + C) + D\), the amplitude is \(|A|\) and the period is \(\frac{2\pi}{B}\). For \(y = 2 \sin x\), A = 2 and B = 1. Therefore, the amplitude is \(|2| = 2\) and the period is \(\frac{2\pi}{1} = 2\pi\) radians or \(360\) degrees.

Identify amplitude and period for y=-4 \cos 2x

For the function \(y = A \cos(Bx + C) + D\), the amplitude is \(|A|\) and the period is \(\frac{2\pi}{B}\). For \(y = -4 \cos 2x\), A = -4 and B = 2. Therefore, the amplitude is \(|-4| = 4\) and the period is \(\frac{2\pi}{2} = \pi\) radians or \(180\) degrees.

Identify amplitude and period for y=\frac{5}{3} \sin \left( -\frac{2}{3} x \right)

For the function \(y = A \sin(Bx + C) + D\), the amplitude is \(|A|\) and the period is \(\frac{2\pi}{|B|}\). For \(y = \frac{5}{3} \sin \left( -\frac{2}{3} x \right)\), A = \(\frac{5}{3}\) and B = \(-\frac{2}{3}\). Therefore, the amplitude is \(\left| \frac{5}{3} \right| = \frac{5}{3}\) and the period is \(\frac{2\pi}{\left| -\frac{2}{3} \right|} = \frac{2\pi}{\frac{2}{3}} = 3\pi\) radians or \(540\) degrees.

Identify amplitude and period for y=3 \cos \frac{1}{2} x

For the function \(y = A \cos(Bx + C) + D\), the amplitude is \(|A|\) and the period is \(\frac{2\pi}{B}\). For \(y = 3 \cos \frac{1}{2} x\), A = 3 and B = \(\frac{1}{2}\). Therefore, the amplitude is \(|3| = 3\) and the period is \(\frac{2\pi}{\frac{1}{2}} = 4\pi\) radians or \(720\) degrees.

Key concepts

Amplitude

The amplitude of a trigonometric function measures how far the function's values oscillate above and below its central axis. For the general functions of the form \(y = A \, \sin(Bx + C) + D\) or \(y = A \, \cos(Bx + C) + D\), the amplitude is given by \(|A|\). This value determines the height of the peaks and the depth of the troughs of the sine or cosine wave.
For example:

• In the function \(y = 2 \, \sin x\), the amplitude is \(|2| = 2\).
• In the function \(y = -4 \, \cos 2x\), the amplitude is \(|-4| = 4\).
• In the function \(y = \frac{5}{3} \, \sin \left( -\frac{2}{3} x \right)\), the amplitude is \(\left| \frac{5}{3} \right| = \frac{5}{3}\).
• In the function \(y = 3 \, \cos \frac{1}{2} x\), the amplitude is \(|3| = 3\).

The larger the amplitude, the taller the graph of the function will be.
Keep this in mind as you analyze or graph trigonometric functions.

Period of a function

The period of a trigonometric function is the distance required for the function to complete one full cycle of its pattern. For the general functions of the form \(y = A \, \sin(Bx + C) + D\) or \(y = A \, \cos(Bx + C) + D\), the period is calculated as \(\frac{2\pi}{|B|}\).
This important characteristic tells you how long it takes for the function to repeat its cycle.
Here are some examples:

• In \(y = 2 \, \sin x\), the period is \(\frac{2\pi}{1} = 2\pi\) radians or 360 degrees.
• In \(y = -4 \, \cos 2x\), the period is \(\frac{2\pi}{2} = \pi\) radians or 180 degrees.
• In \(y = \frac{5}{3} \, \sin \left( -\frac{2}{3} x \right)\), the period is \( \frac{2\pi}{\left| -\frac{2}{3} \right|} = \frac{2\pi}{\frac{2}{3}} = 3\pi\) radians or 540 degrees.
• In \(y = 3 \, \cos \frac{1}{2} x\), the period is \(\frac{2\pi}{\frac{1}{2}} = 4\pi\) radians or 720 degrees.

Understanding the period helps in graphing and analyzing repeating behaviors in real-world scenarios.
Always note the effect of coefficient \(B\) on the period of the function.

Radians and Degrees

Radians and degrees are two units used to measure angles. When working with trigonometric functions, it's essential to be comfortable converting between these units.
A full circle is \(360\) degrees, which is equivalent to \(2\pi\) radians.
Here are a few important conversions to remember:

• \(180\) degrees is equal to \(\pi\) radians.
• To convert from degrees to radians, use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).
• To convert from radians to degrees, use the formula: \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \).

Let’s see how these conversions work in practice:

• If the period of a trigonometric function is \(2\pi\) radians, you can convert it to degrees: \(2\pi \times \frac{180}{\pi} = 360\) degrees.
• If the period is \(\pi\) radians, it converts to \(\pi \times \frac{180}{\pi} = 180\) degrees.
• If you have \(4\pi\) radians, this translates to \(4\pi \times \frac{180}{\pi} = 720\) degrees.

Familiarity with both units and the ease of conversion can help you better understand and communicate results related to trigonometric functions.