Functions that relate the angles of a triangle to the ratios of its sides. The primary trigonometric functions are sine, cosine, and tangent.
Trigonometric functions are powerful tools for modelling real-world scenarios that involve periodic behavior. These functions can be used to represent situations where a quantity repeats itself over a fixed interval of time or space.
Examples of periodic phenomena:
- The height of a point on a Ferris wheel as it rotates.
- The oscillation of a spring or pendulum.
- The variation of daylight hours throughout the year.
- The fluctuation of temperatures over seasons.
In this section, we will explore how to use trigonometric functions to model such scenarios, focusing on the sine and cosine functions.
Modelling with Sine and Cosine Functions
The sine and cosine functions are ideal for modelling periodic behavior because they have the following properties:
- They are periodic with a period of \$2\pi\$ (or \$360^\circ\$).
- They oscillate between a maximum and minimum value, making them suitable for representing cyclical patterns.
The general form of a sine or cosine function is:
\$\$f(x) = a \sin(b(x - c)) + d\$\$
or
\$\$f(x) = a \cos(b(x - c)) + d\$\$
where:
- \$a\$ is the amplitude, which determines the vertical stretch or compression of the function.
- \$b\$ affects the period of the function, which is given by \$\frac{2\pi}{b}\$.
- \$c\$ is the horizontal shift, which moves the function left or right.
- \$d\$ is the vertical shift, which moves the function up or down.
The choice between sine and cosine depends on the starting point of the scenario:
- Use sine if the scenario starts at the midline (average value).
- Use cosine if the scenario starts at a maximum or minimum.
Example: Modelling the Height of a Ferris Wheel
Consider a Ferris wheel with the following characteristics:
- Radius: 10 meters
- Height of the center above the ground: 15 meters
- Time for a complete revolution: 60 seconds
We want to model the height of a point on the Ferris wheel as a function of time.
Step 1: Determine the Amplitude
The amplitude \$a\$ is the distance from the midline to the maximum or minimum value. In this case, the amplitude is equal to the radius of the Ferris wheel:
\$\$a = 10\$\$
Step 2: Determine the Period
The period is the time it takes for the function to complete one cycle. The period of a sine or cosine function is given by \$\frac{2\pi}{b}\$, where \$b\$ is the frequency. Since the Ferris wheel takes 60 seconds for a complete revolution, we have:
\$\$\frac{2\pi}{b} = 60 \implies b = \frac{2\pi}{60} = \frac{\pi}{30}\$\$
Step 3: Determine the Vertical Shift
The vertical shift \$d\$ is the midline of the function. In this case, the midline is the height of the center of the Ferris wheel above the ground:
\$\$d = 15\$\$
Step 4: Determine the Horizontal Shift
The horizontal shift \$c\$ depends on the starting point of the scenario. If the point starts at the bottom of the Ferris wheel, we can use a cosine function with a phase shift of \$\pi\$ (since \$\cos(\pi) = -1\$):
\$\$c = 0\$\$
Step 5: Write the Function
Using the information above, we can write the function for the height of the point on the Ferris wheel as:
\$\$h(t) = 10 \cos\left(\frac{\pi}{30} t + \pi\right) + 15\$\$
Example: Modelling the Temperature Over a Year
Consider the average temperature in a city over a year, which varies between 10°C in winter and 30°C in summer. We want to model the temperature as a function of time, where \$t\$ is the number of months since January.
Step 1: Determine the Amplitude
The amplitude is half the difference between the maximum and minimum values:
\$\$a = \frac{30 - 10}{2} = 10\$\$
Step 2: Determine the Period
The period is 12 months, so:
\$\$\frac{2\pi}{b} = 12 \implies b = \frac{\pi}{6}\$\$
Step 3: Determine the Vertical Shift
The vertical shift is the average of the maximum and minimum values:
\$\$d = \frac{30 + 10}{2} = 20\$\$
Step 4: Determine the Horizontal Shift
Assume the maximum temperature occurs in July (month 6). We can use a cosine function with a phase shift of 6 months:
\$\$c = 6\$\$
Step 5: Write the Function
The function for the temperature is:
\$\$T(t) = 10 \cos\left(\frac{\pi}{6} (t - 6)\right) + 20\$\$
- Try modelling the height of a point on a Ferris wheel with different starting points (e.g., starting at the top or midline). How does the function change?
- Model the daylight hours in a city over a year. What is the amplitude, period, and vertical shift?
How do mathematicians decide which function best models a real-world scenario? What are the limitations of using mathematical models to represent complex phenomena?
