QUESTION 7 Two trig functions namely, $$ f(x)=a \cos x $$ and $$ g(x)=\sin b x $$, are shown below for $$ x \in\left[-180^{\circ} ; 180^{\circ}\right] $$. 7.1 Determine; 7.1 .1 the value of $$ a $$ and $$ b $$. 713 the range of $$ f $$.

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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 $$ b $$.

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### **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:
   
   $$
   \text{Range of } f(x) = [ -3, \, 3 ]
   $$

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### **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.
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 $$ a $$.

QUESTION 7 Two trig functions namely, \( f(x)=a \cos x \) and \( g(x)=\sin b x \), are shown below for \( x \in\left[-180^{\circ} ; 180^{\circ}\right] \). 7.1 Determine; 7.1 .1 the value of \( a \) and \( b \). 713 the range of \( f \).

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