To graph the function [tex]\( y = \sin(0.25x) \)[/tex] where [tex]\( x \)[/tex] is measured in degrees, we can proceed with the following steps:
### 1. Understand the Function
The sine function, [tex]\( \sin(\theta) \)[/tex], typically has a period of [tex]\( 360^\circ \)[/tex]. This means that it repeats itself every [tex]\( 360^\circ \)[/tex]. The argument of the sine function in this case is [tex]\( 0.25x \)[/tex], which means that the period of our function will be different.
### 2. Determine the Period
The period of the function [tex]\( y = \sin(kx) \)[/tex] is given by [tex]\( \frac{360^\circ}{|k|} \)[/tex]. For [tex]\( y = \sin(0.25x) \)[/tex]:
[tex]\[ k = 0.25 \][/tex]
[tex]\[ \text{Period} = \frac{360^\circ}{0.25} = 1440^\circ \][/tex]
Thus, the function [tex]\( y = \sin(0.25x) \)[/tex] will repeat every [tex]\( 1440^\circ \)[/tex].
### 3. Key Points within One Period
To plot the graph accurately, let's calculate some key points:
- When [tex]\( x = 0^\circ \)[/tex]:
[tex]\[ y = \sin(0.25 \times 0) = \sin(0) = 0 \][/tex]
- When [tex]\( x = 360^\circ \)[/tex]:
[tex]\[ y = \sin(0.25 \times 360) = \sin(90^\circ) = 1 \][/tex]
- When [tex]\( x = 720^\circ \)[/tex]:
[tex]\[ y = \sin(0.25 \times 720) = \sin(180^\circ) = 0 \][/tex]
- When [tex]\( x = 1080^\circ \)[/tex]:
[tex]\[ y = \sin(0.25 \times 1080) = \sin(270^\circ) = -1 \][/tex]
- When [tex]\( x = 1440^\circ \)[/tex]:
[tex]\[ y = \sin(0.25 \times 1440) = \sin(360^\circ) = 0 \][/tex]
### 4. Plotting the Graph
These points give us enough information to sketch one complete cycle of the function [tex]\( y = \sin(0.25x) \)[/tex]:
- The function starts at the origin [tex]\( (0^\circ, 0) \)[/tex].
- It increases to a maximum of 1 at [tex]\( 360^\circ \)[/tex].
- It returns to 0 at [tex]\( 720^\circ \)[/tex].
- It reaches a minimum of -1 at [tex]\( 1080^\circ \)[/tex].
- It comes back to 0 at [tex]\( 1440^\circ \)[/tex].
Between these key points, the function will follow the smooth, wave-like pattern characteristic of the sine function.
### 5. Sketch the Graph
To visualize the graph over one period (from [tex]\( 0^\circ \)[/tex] to [tex]\( 1440^\circ \)[/tex]):
- The graph should consist of a sinusoidal wave starting at (0,0), reaching the peak at [tex]\( 360^\circ \)[/tex], crossing back through (720,0), reaching the trough at [tex]\( 1080^\circ \)[/tex], and finally completing the cycle back at (1440,0).
### Summary
The graph of [tex]\( y = \sin(0.25x) \)[/tex] in degrees will display a sine wave with a period of [tex]\( 1440^\circ \)[/tex], and the key points will be:
1. [tex]\( (0, 0) \)[/tex]
2. [tex]\( (360, 1) \)[/tex]
3. [tex]\( (720, 0) \)[/tex]
4. [tex]\( (1080, -1) \)[/tex]
5. [tex]\( (1440, 0) \)[/tex]
This comprehensive step-by-step explanation should help you understand the graph of [tex]\( y = \sin(0.25x) \)[/tex] over the specified range.
