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To find the values of \( a \) and \( b \), measure the amplitude of \( f(x) = a \cos x \) to determine \( a \), and calculate \( b \) using the period of \( g(x) = \sin(bx) \). The range of \( f(x) \) is from \( -a \) to \( a \).
Solución
To determine the values of \( a \) and \( b \) for the given trigonometric functions and to find the range of \( f(x) \), we'll analyze each part step by step.
### **Given Functions:**
1. \( f(x) = a \cos x \)
2. \( g(x) = \sin(bx) \)
### **Assumptions:**
Since the graphs are provided for \( x \in [-180^\circ, 180^\circ] \), we'll use the standard properties of trigonometric functions to determine the values of \( a \) and \( b \).
---
### **7.1.1 Determine the Values of \( a \) and \( b \)**
#### **Determining \( a \) in \( f(x) = a \cos x \):**
1. **Amplitude Analysis:**
- The general form of the cosine function is \( A \cos(Bx + C) + D \), where:
- \( A \) is the amplitude.
- \( B \) affects the period.
- \( C \) is the phase shift.
- \( D \) is the vertical shift.
- For \( f(x) = a \cos x \):
- **Amplitude (\( a \))**: This is the coefficient multiplying the cosine function. It determines the peak value (maximum and minimum) of the function.
2. **From the Graph:**
- **Identify the Maximum and Minimum Values** of \( f(x) \) from the graph.
- The **Amplitude \( a \)** is the absolute value of the maximum (or minimum) since cosine functions are symmetric.
\[
a = \text{Maximum value of } |f(x)|
\]
- **Example:** If the graph of \( f(x) \) reaches a maximum of 3 and a minimum of -3, then \( a = 3 \).
#### **Determining \( b \) in \( g(x) = \sin(bx) \):**
1. **Period Analysis:**
- The general period of the sine function \( \sin(Bx) \) is given by:
\[
\text{Period} = \frac{360^\circ}{|B|}
\]
- For \( g(x) = \sin(bx) \):
- \( B = b \), so:
\[
\text{Period of } g(x) = \frac{360^\circ}{b}
\]
2. **From the Graph:**
- **Determine the Period** of \( g(x) \) by measuring the distance between two consecutive peaks (or troughs).
- **Calculate \( b \)** using the period.
\[
b = \frac{360^\circ}{\text{Period}}
\]
- **Example:** If \( g(x) \) completes two full cycles within \( 360^\circ \), then the period is \( 180^\circ \), and:
\[
b = \frac{360^\circ}{180^\circ} = 2
\]
---
### **7.1.3 Determine the Range of \( f(x) = a \cos x \)**
1. **Understanding Range:**
- The range of a function is the set of all possible output values (y-values).
2. **For \( f(x) = a \cos x \):**
- The cosine function oscillates between -1 and 1.
- Multiplying by \( a \) scales this range.
\[
\text{Range of } f(x) = [ -a, \, a ]
\]
3. **Example:**
- If \( a = 3 \), then:
\[
\text{Range of } f(x) = [ -3, \, 3 ]
\]
---
### **Summary:**
- **\( a \)** is determined by the **amplitude** of the \( \cos x \) graph.
- **\( b \)** is calculated based on the **period** of the \( \sin(bx) \) graph.
- The **range of \( f(x) \)** is from **\( -a \) to \( a \)**.
**Note:** To obtain specific numerical values for \( a \) and \( b \), you would need to refer to the actual graphs of \( f(x) \) and \( g(x) \). Measure the amplitude from \( f(x) = a \cos x \) and determine the period of \( g(x) = \sin(bx) \) to calculate \( b \) accordingly.
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