Answer :
Let's solve the problem step by step.
### Step 1: Define the Range of [tex]\( x \)[/tex]
We need to plot the functions for [tex]\( 0^\circ \leq x \leq 360^\circ \)[/tex]. This means [tex]\( x \)[/tex] ranges from 0 to 360 degrees.
### Step 2: Define the Functions
We have two functions:
[tex]\[ f(x) = 4\cos(x) \][/tex]
[tex]\[ g(x) = 4\sin(x) \][/tex]
These are trigonometric functions, where [tex]\( f(x) \)[/tex] represents a scaled cosine function and [tex]\( g(x) \)[/tex] represents a scaled sine function.
### Step 3: Identifying Key Points and Intercepts
To plot these functions, it's crucial to identify key points and intercepts with the x-axis.
#### Intercepts with the x-axis
- For [tex]\( f(x) = 4\cos(x) \)[/tex]:
The cosine function equals zero at [tex]\( x = 90^\circ \)[/tex] and [tex]\( x = 270^\circ \)[/tex].
So, the intercepts are:
[tex]\[ f(90^\circ) = 4\cos(90^\circ) = 4 \times 0 = 0 \][/tex]
[tex]\[ f(270^\circ) = 4\cos(270^\circ) = 4 \times 0 = 0 \][/tex]
- For [tex]\( g(x) = 4\sin(x) \)[/tex]:
The sine function equals zero at [tex]\( x = 0^\circ \)[/tex], [tex]\( x = 180^\circ \)[/tex], and [tex]\( x = 360^\circ \)[/tex].
So, the intercepts are:
[tex]\[ g(0^\circ) = 4\sin(0^\circ) = 4 \times 0 = 0 \][/tex]
[tex]\[ g(180^\circ) = 4\sin(180^\circ) = 4 \times 0 = 0 \][/tex]
[tex]\[ g(360^\circ) = 4\sin(360^\circ) = 4 \times 0 = 0 \][/tex]
### Step 4: Plot the Functions
To plot [tex]\( f(x) = 4\cos(x) \)[/tex] and [tex]\( g(x) = 4\sin(x) \)[/tex], you can follow these steps:
1. Set up the coordinate system:
- The x-axis ranges from 0 to 360 degrees.
- The y-axis ranges from -4 to 4 (since the amplitudes of both functions are 4).
2. Plot [tex]\( f(x) = 4\cos(x) \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 4\cos(0) = 4 \)[/tex].
- At [tex]\( x = 90 \)[/tex], [tex]\( f(90) = 4\cos(90) = 0 \)[/tex].
- At [tex]\( x = 180 \)[/tex], [tex]\( f(180) = 4\cos(180) = -4 \)[/tex].
- At [tex]\( x = 270 \)[/tex], [tex]\( f(270) = 4\cos(270) = 0 \)[/tex].
- At [tex]\( x = 360 \)[/tex], [tex]\( f(360) = 4\cos(360) = 4 \)[/tex].
3. Plot [tex]\( g(x) = 4\sin(x) \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( g(0) = 4\sin(0) = 0 \)[/tex].
- At [tex]\( x = 90 \)[/tex], [tex]\( g(90) = 4\sin(90) = 4 \)[/tex].
- At [tex]\( x = 180 \)[/tex], [tex]\( g(180) = 4\sin(180) = 0 \)[/tex].
- At [tex]\( x = 270 \)[/tex], [tex]\( g(270) = 4\sin(270) = -4 \)[/tex].
- At [tex]\( x = 360 \)[/tex], [tex]\( g(360) = 4\sin(360) = 0 \)[/tex].
### Step 5: Drawing the Graphs
1. Draw [tex]\( f(x) = 4\cos(x) \)[/tex]:
- Start at (0, 4), go through (90, 0), (180, -4), (270, 0), and back to (360, 4).
- This will create a cosine wave that oscillates between 4 and -4.
2. Draw [tex]\( g(x) = 4\sin(x) \)[/tex]:
- Start at (0, 0), go through (90, 4), (180, 0), (270, -4), and back to (360, 0).
- This will create a sine wave that also oscillates between 4 and -4.
### Final Intercepts on the x-axis:
- For [tex]\( f(x) = 4\cos(x) \)[/tex]: Intercepts at [tex]\( 90^\circ \)[/tex] and [tex]\( 270^\circ \)[/tex].
- For [tex]\( g(x) = 4\sin(x) \)[/tex]: Intercepts at [tex]\( 0^\circ \)[/tex], [tex]\( 180^\circ \)[/tex], and [tex]\( 360^\circ \)[/tex].
### Conclusion
The graphs of [tex]\( f(x) = 4\cos(x) \)[/tex] and [tex]\( g(x) = 4\sin(x) \)[/tex] intersect the x-axis at different points. Plotting these will display their waveforms oscillating between 4 and -4 within the given range.
### Step 1: Define the Range of [tex]\( x \)[/tex]
We need to plot the functions for [tex]\( 0^\circ \leq x \leq 360^\circ \)[/tex]. This means [tex]\( x \)[/tex] ranges from 0 to 360 degrees.
### Step 2: Define the Functions
We have two functions:
[tex]\[ f(x) = 4\cos(x) \][/tex]
[tex]\[ g(x) = 4\sin(x) \][/tex]
These are trigonometric functions, where [tex]\( f(x) \)[/tex] represents a scaled cosine function and [tex]\( g(x) \)[/tex] represents a scaled sine function.
### Step 3: Identifying Key Points and Intercepts
To plot these functions, it's crucial to identify key points and intercepts with the x-axis.
#### Intercepts with the x-axis
- For [tex]\( f(x) = 4\cos(x) \)[/tex]:
The cosine function equals zero at [tex]\( x = 90^\circ \)[/tex] and [tex]\( x = 270^\circ \)[/tex].
So, the intercepts are:
[tex]\[ f(90^\circ) = 4\cos(90^\circ) = 4 \times 0 = 0 \][/tex]
[tex]\[ f(270^\circ) = 4\cos(270^\circ) = 4 \times 0 = 0 \][/tex]
- For [tex]\( g(x) = 4\sin(x) \)[/tex]:
The sine function equals zero at [tex]\( x = 0^\circ \)[/tex], [tex]\( x = 180^\circ \)[/tex], and [tex]\( x = 360^\circ \)[/tex].
So, the intercepts are:
[tex]\[ g(0^\circ) = 4\sin(0^\circ) = 4 \times 0 = 0 \][/tex]
[tex]\[ g(180^\circ) = 4\sin(180^\circ) = 4 \times 0 = 0 \][/tex]
[tex]\[ g(360^\circ) = 4\sin(360^\circ) = 4 \times 0 = 0 \][/tex]
### Step 4: Plot the Functions
To plot [tex]\( f(x) = 4\cos(x) \)[/tex] and [tex]\( g(x) = 4\sin(x) \)[/tex], you can follow these steps:
1. Set up the coordinate system:
- The x-axis ranges from 0 to 360 degrees.
- The y-axis ranges from -4 to 4 (since the amplitudes of both functions are 4).
2. Plot [tex]\( f(x) = 4\cos(x) \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 4\cos(0) = 4 \)[/tex].
- At [tex]\( x = 90 \)[/tex], [tex]\( f(90) = 4\cos(90) = 0 \)[/tex].
- At [tex]\( x = 180 \)[/tex], [tex]\( f(180) = 4\cos(180) = -4 \)[/tex].
- At [tex]\( x = 270 \)[/tex], [tex]\( f(270) = 4\cos(270) = 0 \)[/tex].
- At [tex]\( x = 360 \)[/tex], [tex]\( f(360) = 4\cos(360) = 4 \)[/tex].
3. Plot [tex]\( g(x) = 4\sin(x) \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( g(0) = 4\sin(0) = 0 \)[/tex].
- At [tex]\( x = 90 \)[/tex], [tex]\( g(90) = 4\sin(90) = 4 \)[/tex].
- At [tex]\( x = 180 \)[/tex], [tex]\( g(180) = 4\sin(180) = 0 \)[/tex].
- At [tex]\( x = 270 \)[/tex], [tex]\( g(270) = 4\sin(270) = -4 \)[/tex].
- At [tex]\( x = 360 \)[/tex], [tex]\( g(360) = 4\sin(360) = 0 \)[/tex].
### Step 5: Drawing the Graphs
1. Draw [tex]\( f(x) = 4\cos(x) \)[/tex]:
- Start at (0, 4), go through (90, 0), (180, -4), (270, 0), and back to (360, 4).
- This will create a cosine wave that oscillates between 4 and -4.
2. Draw [tex]\( g(x) = 4\sin(x) \)[/tex]:
- Start at (0, 0), go through (90, 4), (180, 0), (270, -4), and back to (360, 0).
- This will create a sine wave that also oscillates between 4 and -4.
### Final Intercepts on the x-axis:
- For [tex]\( f(x) = 4\cos(x) \)[/tex]: Intercepts at [tex]\( 90^\circ \)[/tex] and [tex]\( 270^\circ \)[/tex].
- For [tex]\( g(x) = 4\sin(x) \)[/tex]: Intercepts at [tex]\( 0^\circ \)[/tex], [tex]\( 180^\circ \)[/tex], and [tex]\( 360^\circ \)[/tex].
### Conclusion
The graphs of [tex]\( f(x) = 4\cos(x) \)[/tex] and [tex]\( g(x) = 4\sin(x) \)[/tex] intersect the x-axis at different points. Plotting these will display their waveforms oscillating between 4 and -4 within the given range.
