91Ó°ÊÓ

The cosine function is a fundamental concept in trigonometry. It helps describe periodic phenomena such as sound waves, light waves, and in this exercise, temperature changes. The general form of a cosine function is given by \( y = a \cos(bx + c) + d \). Here is a breakdown of the components: - **Amplitude (\( a \))**: This dictates how high and low the wave reaches from its center point. - **Frequency/B-value (\( b \))**: Affects the number of cycles within a given interval. - **Phase shift/C-value (\( c \))**: Determines horizontal shifting along the x-axis. - **Vertical shift/D-value (\( d \))**: Moves the function up or down the y-axis.
In this exercise, the cosine function helps model the outdoor temperature over time: - The equation is \( y = 80 + 22 \cos\left[ \frac{\pi}{12}(t-3) \right] \). - Temperature fluctuates around a mean value (80°F), showcasing the natural cycle depicted by the changes in the wave via the cosine function.

Amplitude and period are two essential characteristics of a cosine function. Understanding them helps interpret models like the one we've seen with temperature changes. **Amplitude** The amplitude refers to the maximum deviation from the mean value in a wave. In the temperature function \( y = 80 + 22 \cos\left[ \frac{\pi}{12}(t-3) \right] \), the amplitude is 22°F. This indicates that the temperature moves 22°F above and below the mean value of 80°F. So, temperatures range between 58°F and 102°F. **Period** Period describes the length of one full wave cycle, showing how long it takes for the function to repeat its behavior. The standard cosine function \( \cos(x) \) has a period of \( 2\pi \). However, the given function's period is influenced by the coefficient \( \frac{\pi}{12} \). Calculating the period: \[ \text{Period} = \frac{2\pi}{\frac{\pi}{12}} = 24 \] Thus, the temperature cycle repeats every 24 hours, highlighting a full daily temperature swing.

Horizontal shift in a trigonometric function occurs when the whole wave is moved along the x-axis. This affects where the wave starts and can significantly change the interpretation of a function in real-world applications.
In our temperature model, \( y = 80 + 22 \cos\left[ \frac{\pi}{12}(t-3) \right] \), the term \( (t-3) \) introduces a horizontal shift. The usual starting point of \( t = 0 \) is adjusted by \( +3 \), moving the start of the cosine wave to \( t = 3 \). This shift implies: - The maximum temperature occurs not at the initial point but 3 units (or hours) later in this 24-hour cycle. - We see the temperature peak at noon instead of 9 A.M., illustrating how a horizontal shift aligns the function closely with real-time changes.