Modeling the same periodic function with sine and cosine
sine vs. cosine
Any periodic phenomenon can be modeled using either a sine or cosine function. The key is understanding that cosine and sine functions are phase shifts of each other:
[latex]\cos(x) = \sin(x + \frac{\pi}{2})[/latex] and [latex]\sin(x) = \cos(x - \frac{\pi}{2})[/latex].
When modeling real-world periodic behavior, you can choose whichever function makes the mathematics simpler or aligns better with your reference point.
Both sine and cosine functions have the same basic shape – they’re both sinusoidal. The only difference is their starting point:
- Cosine starts at its maximum value when the input is 0
- Sine starts at its middle value (crossing the axis) when the input is 0
Suppose the temperature in Phoenix, Arizona follows this pattern:
- Maximum temperature: 85°F at 3:00 PM ([latex]t = 15[/latex] hours)
- Minimum temperature: 65°F at 3:00 AM ([latex]t = 3[/latex] hours)
- The pattern repeats every 24 hours
Let’s find both sine and cosine functions to model this behavior.
Step 1: Identify the parameters
- Amplitude: [latex]A = \frac{85 - 65}{2} = 10[/latex]
- Midline: [latex]D = \frac{85 + 65}{2} = 75[/latex]
- Period = 24, so [latex]B = \frac{2\pi}{24} = \frac{\pi}{12}[/latex]
Step 2: Write the cosine function
Since cosine starts at its maximum, and our maximum occurs at [latex]t = 15[/latex]: [latex]T(t) = 10 \cos\left(\frac{\pi}{12}(t - 15)\right) + 75[/latex]
Step 3: Write the equivalent sine function
For sine to reach maximum at [latex]t = 15[/latex], we need: [latex]\sin\left(\frac{\pi}{12}(t - C)\right) = 1[/latex] This happens when [latex]\frac{\pi}{12}(t - C) = \frac{\pi}{2}[/latex] So: [latex]t - C = 6[/latex], which means [latex]C = 15 - 6 = 9[/latex]
[latex]T(t) = 10 \sin\left(\frac{\pi}{12}(t - 9)\right) + 75[/latex]
Verification: Both functions give [latex]T(15) = 85°F[/latex] and [latex]T(3) = 65°F[/latex]
Question Help: Converting Between Sine and Cosine
When you have one trigonometric function and need to write the equivalent using the other:
- From cosine to sine: Replace cos with sin and adjust the phase shift by subtracting [latex]\frac{1}{4}[/latex] the period from the argument
- From sine to cosine: Replace sin with cos and adjust the phase shift by adding latex]\frac{1}{4}[/latex] to the argument
- Check your work: Both functions should give the same values at key points (maximum, minimum, zeros)
A Ferris wheel has a diameter of 50 feet with its center 30 feet above the ground. The wheel completes one revolution every 8 minutes. If a rider starts at the bottom of the wheel when t = 0:
- Write a cosine function h(t) to model the rider’s height above ground.
- Write an equivalent sine function for the same motion.
Answer: [latex]h(t) = -25 \cos\left(\frac{\pi}{4}t\right) + 30[/latex] and [latex]h(t) = 25 \sin\left(\frac{\pi}{4}(t - 2)\right) + 30[/latex]
- Use cosine if your phenomenon starts at a maximum or minimum value
- Use sine if your phenomenon starts at the middle value (crossing the midline)
This choice can make your phase shift calculation much simpler!
A sound wave oscillates with amplitude 0.5 units, frequency 440 Hz glossary: hertz, a unit measuring cycles per second, and passes through the origin [latex](0,0)[/latex] while increasing.