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To graph the function [tex]\(y = \cos \left( \frac{3\pi}{4} \right) \sin (x) + \cos (x) \sin \left( \frac{3\pi}{4} \right)\)[/tex] over the interval [tex]\( [-2\pi, 2\pi] \)[/tex], we need to first rewrite it using trigonometric identities. Here's the detailed, step-by-step solution:
1. Use the Angle Sum Identity for Sine:
The given function is of the form [tex]\( \cos(A) \sin(x) + \cos(x) \sin(A) \)[/tex]. Using the sum identity for sine, we know:
[tex]\[ \sin(A + x) = \sin(x) \cos(A) + \cos(x) \sin(A) \][/tex]
2. Match the Given Expression:
By comparing the given function to the sine sum identity, it is clear that:
[tex]\[ y = \cos \left( \frac{3\pi}{4} \right) \sin (x) + \cos (x) \sin \left( \frac{3\pi}{4} \right) \][/tex]
corresponds to:
[tex]\[ y = \sin \left( x + \frac{3\pi}{4} \right) \][/tex]
3. Simplify Using Trigonometric Values:
Let's verify the trigonometric values used in the identity:
- [tex]\( \cos \left( \frac{3\pi}{4} \right) = -\frac{\sqrt{2}}{2} \)[/tex]
- [tex]\( \sin \left( \frac{3\pi}{4} \right) = \frac{\sqrt{2}}{2} \)[/tex]
Therefore, we indeed have:
[tex]\[ y = \sin \left( x + \frac{3\pi}{4} \right) \][/tex]
4. Graphing the Function:
Now that we have the function in a simpler form, [tex]\( y = \sin \left( x + \frac{3\pi}{4} \right) \)[/tex], we can proceed to graph it. Here's how:
- Original Sine Function: Recall that the basic [tex]\( y = \sin(x) \)[/tex] function oscillates between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex] with a period of [tex]\(2\pi\)[/tex], crossing the x-axis at [tex]\(0, \pi, 2\pi\)[/tex] (and their negative counterparts).
- Phase Shift: The function [tex]\( y = \sin \left( x + \frac{3\pi}{4} \right) \)[/tex] is a horizontal shift of [tex]\( y = \sin(x) \)[/tex] to the left by [tex]\( \frac{3\pi}{4} \)[/tex].
- Interval [tex]\([-2\pi, 2\pi]\)[/tex]:
- For [tex]\( x = -2\pi \)[/tex]:
[tex]\[ y = \sin \left( -2\pi + \frac{3\pi}{4} \right) = \sin \left( -\frac{5\pi}{4} \right) \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \sin \left( 0 + \frac{3\pi}{4} \right) = \sin \left( \frac{3\pi}{4} \right) = \frac{\sqrt{2}}{2} \][/tex]
- For [tex]\( x = 2\pi \)[/tex]:
[tex]\[ y = \sin \left( 2\pi + \frac{3\pi}{4} \right) = \sin \left( \frac{11\pi}{4} \right) \][/tex]
5. Sketching the Graph:
- Draw the [tex]\( y = \sin(x) \)[/tex] curve.
- Shift this curve to the left by [tex]\( \frac{3\pi}{4} \)[/tex].
The resultant graph should show a sine wave starting at [tex]\( -2\pi \)[/tex] and ending at [tex]\( 2\pi \)[/tex], shifted to reflect the phase shift of [tex]\( \frac{3\pi}{4} \)[/tex]. The values and crossings will all be adjusted according to this shift.
This completes the step-by-step explanation of how to rewrite and graph the given trigonometric function.
1. Use the Angle Sum Identity for Sine:
The given function is of the form [tex]\( \cos(A) \sin(x) + \cos(x) \sin(A) \)[/tex]. Using the sum identity for sine, we know:
[tex]\[ \sin(A + x) = \sin(x) \cos(A) + \cos(x) \sin(A) \][/tex]
2. Match the Given Expression:
By comparing the given function to the sine sum identity, it is clear that:
[tex]\[ y = \cos \left( \frac{3\pi}{4} \right) \sin (x) + \cos (x) \sin \left( \frac{3\pi}{4} \right) \][/tex]
corresponds to:
[tex]\[ y = \sin \left( x + \frac{3\pi}{4} \right) \][/tex]
3. Simplify Using Trigonometric Values:
Let's verify the trigonometric values used in the identity:
- [tex]\( \cos \left( \frac{3\pi}{4} \right) = -\frac{\sqrt{2}}{2} \)[/tex]
- [tex]\( \sin \left( \frac{3\pi}{4} \right) = \frac{\sqrt{2}}{2} \)[/tex]
Therefore, we indeed have:
[tex]\[ y = \sin \left( x + \frac{3\pi}{4} \right) \][/tex]
4. Graphing the Function:
Now that we have the function in a simpler form, [tex]\( y = \sin \left( x + \frac{3\pi}{4} \right) \)[/tex], we can proceed to graph it. Here's how:
- Original Sine Function: Recall that the basic [tex]\( y = \sin(x) \)[/tex] function oscillates between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex] with a period of [tex]\(2\pi\)[/tex], crossing the x-axis at [tex]\(0, \pi, 2\pi\)[/tex] (and their negative counterparts).
- Phase Shift: The function [tex]\( y = \sin \left( x + \frac{3\pi}{4} \right) \)[/tex] is a horizontal shift of [tex]\( y = \sin(x) \)[/tex] to the left by [tex]\( \frac{3\pi}{4} \)[/tex].
- Interval [tex]\([-2\pi, 2\pi]\)[/tex]:
- For [tex]\( x = -2\pi \)[/tex]:
[tex]\[ y = \sin \left( -2\pi + \frac{3\pi}{4} \right) = \sin \left( -\frac{5\pi}{4} \right) \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = \sin \left( 0 + \frac{3\pi}{4} \right) = \sin \left( \frac{3\pi}{4} \right) = \frac{\sqrt{2}}{2} \][/tex]
- For [tex]\( x = 2\pi \)[/tex]:
[tex]\[ y = \sin \left( 2\pi + \frac{3\pi}{4} \right) = \sin \left( \frac{11\pi}{4} \right) \][/tex]
5. Sketching the Graph:
- Draw the [tex]\( y = \sin(x) \)[/tex] curve.
- Shift this curve to the left by [tex]\( \frac{3\pi}{4} \)[/tex].
The resultant graph should show a sine wave starting at [tex]\( -2\pi \)[/tex] and ending at [tex]\( 2\pi \)[/tex], shifted to reflect the phase shift of [tex]\( \frac{3\pi}{4} \)[/tex]. The values and crossings will all be adjusted according to this shift.
This completes the step-by-step explanation of how to rewrite and graph the given trigonometric function.
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