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Sagot :
To solve this problem, we will follow the instructions step-by-step:
### Part (a)
We need to use the product-to-sum formulas to express the product [tex]\( H(x) = 2 \sin (2x) \cos (4x) \)[/tex] as a sum.
The product-to-sum identities tell us:
[tex]\[ 2 \sin A \cos B = \sin(A + B) + \sin(A - B) \][/tex]
Here, [tex]\( A = 2x \)[/tex] and [tex]\( B = 4x \)[/tex].
Applying the identity:
[tex]\[ 2 \sin(2x) \cos(4x) = \sin(2x + 4x) + \sin(2x - 4x) \][/tex]
Simplify the arguments of the sine functions:
[tex]\[ \sin(2x + 4x) + \sin(2x - 4x) = \sin(6x) + \sin(-2x) \][/tex]
Using the fact that [tex]\( \sin(-\theta) = -\sin(\theta) \)[/tex]:
[tex]\[ \sin(6x) + \sin(-2x) = \sin(6x) - \sin(2x) \][/tex]
Therefore, we have:
[tex]\[ H(x) = 2 \sin (2x) \cos (4x) = \sin (6x) - \sin (2x) \][/tex]
So, the simplified form is:
[tex]\[ H(x) = \boxed{\sin(6x) - \sin(2x)} \][/tex]
### Part (b)
We now want to graph the function [tex]\( H(x) = \sin(6x) - \sin(2x) \)[/tex] over the interval [tex]\([0, 2\pi]\)[/tex]. To do this, we'll calculate and plot values of [tex]\( H(x) \)[/tex] at specific points within this interval. Common points for graphing trigonometric functions are multiples of [tex]\(\pi/6\)[/tex].
Let's calculate [tex]\( H(x) \)[/tex] at these points:
[tex]\[ \begin{align*} H(0) &= \sin(6 \cdot 0) - \sin(2 \cdot 0) = 0, \\ H\left( \frac{\pi}{6} \right) &= \sin\left( 6 \cdot \frac{\pi}{6} \right) - \sin\left( 2 \cdot \frac{\pi}{6} \right) = \sin(\pi) - \sin\left( \frac{\pi}{3} \right) = 0 - \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2}, \\ H\left( \frac{\pi}{3} \right) &= \sin\left( 6 \cdot \frac{\pi}{3} \right) - \sin\left( 2 \cdot \frac{\pi}{3} \right) = \sin(2\pi) - \sin\left( \frac{2\pi}{3} \right) = 0 - \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2}, \\ H\left( \frac{\pi}{2} \right) &= \sin\left( 6 \cdot \frac{\pi}{2} \right) - \sin\left( 2 \cdot \frac{\pi}{2} \right) = \sin(3\pi) - \sin(\pi) = 0 - 0 = 0, \\ H\left( \frac{2\pi}{3} \right) &= \sin\left( 6 \cdot \frac{2\pi}{3} \right) - \sin\left( 2 \cdot \frac{2\pi}{3} \right) = \sin(4\pi) - \sin(\frac{4\pi}{3}) = 0 + \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}, \\ H\left( \frac{5\pi}{6} \right) &= \sin\left( 6 \cdot \frac{5\pi}{6} \right) - \sin\left( 2 \cdot \frac{5\pi}{6} \right) = \sin(5\pi) - \sin\left( \frac{5\pi}{3} \right) = 0 + \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}, \\ H(\pi) &= \sin(6 \pi) - \sin(2 \pi) = 0 - 0 = 0, \\ H\left( \frac{7\pi}{6} \right) &= \sin\left( 6 \cdot \frac{7\pi}{6} \right) - \sin\left( 2 \cdot \frac{7\pi}{6} \right) = \sin(7\pi) - \sin\left( \frac{7\pi}{3} \right) = 0 - \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2}, \\ H\left( \frac{4\pi}{3} \right) &= \sin\left( 6 \cdot \frac{4\pi}{3} \right) - \sin\left( 2 \cdot \frac{4\pi}{3} \right) = \sin(8\pi) - \sin\left( \frac{8\pi}{3} \right) = 0 - \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2}, \\ H\left( \frac{3\pi}{2} \right) &= \sin\left( 6 \cdot \frac{3\pi}{2} \right) - \sin\left( 2 \cdot \frac{3\pi}{2} \right) = \sin(9\pi) - \sin(3\pi) = 0 - 0 = 0, \\ H\left( \frac{5\pi}{3} \right) &= \sin\left( 6 \cdot \frac{5\pi}{3} \right) - \sin\left( 2 \cdot \frac{5\pi}{3} \right) = \sin(10\pi) - \sin\left( \frac{10\pi}{3} \right) = 0 + \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}, \\ H\left( \frac{11\pi}{6} \right) &= \sin\left( 6 \cdot \frac{11\pi}{6} \right) - \sin\left( 2 \cdot \frac{11\pi}{6} \right) = \sin(11\pi) - \sin\left( \frac{11\pi}{3} \right) = 0 + \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}, \\ H(2\pi) &= \sin(12\pi) - \sin(4\pi) = 0 - 0 = 0. \end{align*} \][/tex]
The calculated values are:
[tex]\[ \begin{array}{c|c} x & H(x) \\ \hline 0 & 0 \\ \frac{\pi}{6} & -\frac{\sqrt{3}}{2} \\ \frac{\pi}{3} & -\frac{\sqrt{3}}{2} \\ \frac{\pi}{2} & 0 \\ \frac{2\pi}{3} & \frac{\sqrt{3}}{2} \\ \frac{5\pi}{6} & \frac{\sqrt{3}}{2} \\ \pi & 0 \\ \frac{7\pi}{6} & -\frac{\sqrt{3}}{2} \\ \frac{4\pi}{3} & -\frac{\sqrt{3}}{2} \\ \frac{3\pi}{2} & 0 \\ \frac{5\pi}{3} & \frac{\sqrt{3}}{2} \\ \frac{11\pi}{6} & \frac{\sqrt{3}}{2} \\ 2\pi & 0 \\ \end{array} \][/tex]
These points can be plotted to create the graph of [tex]\( H(x) = \sin(6x) - \sin(2x) \)[/tex] over the interval [tex]\([0, 2\pi]\)[/tex].
### Part (a)
We need to use the product-to-sum formulas to express the product [tex]\( H(x) = 2 \sin (2x) \cos (4x) \)[/tex] as a sum.
The product-to-sum identities tell us:
[tex]\[ 2 \sin A \cos B = \sin(A + B) + \sin(A - B) \][/tex]
Here, [tex]\( A = 2x \)[/tex] and [tex]\( B = 4x \)[/tex].
Applying the identity:
[tex]\[ 2 \sin(2x) \cos(4x) = \sin(2x + 4x) + \sin(2x - 4x) \][/tex]
Simplify the arguments of the sine functions:
[tex]\[ \sin(2x + 4x) + \sin(2x - 4x) = \sin(6x) + \sin(-2x) \][/tex]
Using the fact that [tex]\( \sin(-\theta) = -\sin(\theta) \)[/tex]:
[tex]\[ \sin(6x) + \sin(-2x) = \sin(6x) - \sin(2x) \][/tex]
Therefore, we have:
[tex]\[ H(x) = 2 \sin (2x) \cos (4x) = \sin (6x) - \sin (2x) \][/tex]
So, the simplified form is:
[tex]\[ H(x) = \boxed{\sin(6x) - \sin(2x)} \][/tex]
### Part (b)
We now want to graph the function [tex]\( H(x) = \sin(6x) - \sin(2x) \)[/tex] over the interval [tex]\([0, 2\pi]\)[/tex]. To do this, we'll calculate and plot values of [tex]\( H(x) \)[/tex] at specific points within this interval. Common points for graphing trigonometric functions are multiples of [tex]\(\pi/6\)[/tex].
Let's calculate [tex]\( H(x) \)[/tex] at these points:
[tex]\[ \begin{align*} H(0) &= \sin(6 \cdot 0) - \sin(2 \cdot 0) = 0, \\ H\left( \frac{\pi}{6} \right) &= \sin\left( 6 \cdot \frac{\pi}{6} \right) - \sin\left( 2 \cdot \frac{\pi}{6} \right) = \sin(\pi) - \sin\left( \frac{\pi}{3} \right) = 0 - \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2}, \\ H\left( \frac{\pi}{3} \right) &= \sin\left( 6 \cdot \frac{\pi}{3} \right) - \sin\left( 2 \cdot \frac{\pi}{3} \right) = \sin(2\pi) - \sin\left( \frac{2\pi}{3} \right) = 0 - \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2}, \\ H\left( \frac{\pi}{2} \right) &= \sin\left( 6 \cdot \frac{\pi}{2} \right) - \sin\left( 2 \cdot \frac{\pi}{2} \right) = \sin(3\pi) - \sin(\pi) = 0 - 0 = 0, \\ H\left( \frac{2\pi}{3} \right) &= \sin\left( 6 \cdot \frac{2\pi}{3} \right) - \sin\left( 2 \cdot \frac{2\pi}{3} \right) = \sin(4\pi) - \sin(\frac{4\pi}{3}) = 0 + \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}, \\ H\left( \frac{5\pi}{6} \right) &= \sin\left( 6 \cdot \frac{5\pi}{6} \right) - \sin\left( 2 \cdot \frac{5\pi}{6} \right) = \sin(5\pi) - \sin\left( \frac{5\pi}{3} \right) = 0 + \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}, \\ H(\pi) &= \sin(6 \pi) - \sin(2 \pi) = 0 - 0 = 0, \\ H\left( \frac{7\pi}{6} \right) &= \sin\left( 6 \cdot \frac{7\pi}{6} \right) - \sin\left( 2 \cdot \frac{7\pi}{6} \right) = \sin(7\pi) - \sin\left( \frac{7\pi}{3} \right) = 0 - \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2}, \\ H\left( \frac{4\pi}{3} \right) &= \sin\left( 6 \cdot \frac{4\pi}{3} \right) - \sin\left( 2 \cdot \frac{4\pi}{3} \right) = \sin(8\pi) - \sin\left( \frac{8\pi}{3} \right) = 0 - \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2}, \\ H\left( \frac{3\pi}{2} \right) &= \sin\left( 6 \cdot \frac{3\pi}{2} \right) - \sin\left( 2 \cdot \frac{3\pi}{2} \right) = \sin(9\pi) - \sin(3\pi) = 0 - 0 = 0, \\ H\left( \frac{5\pi}{3} \right) &= \sin\left( 6 \cdot \frac{5\pi}{3} \right) - \sin\left( 2 \cdot \frac{5\pi}{3} \right) = \sin(10\pi) - \sin\left( \frac{10\pi}{3} \right) = 0 + \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}, \\ H\left( \frac{11\pi}{6} \right) &= \sin\left( 6 \cdot \frac{11\pi}{6} \right) - \sin\left( 2 \cdot \frac{11\pi}{6} \right) = \sin(11\pi) - \sin\left( \frac{11\pi}{3} \right) = 0 + \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}, \\ H(2\pi) &= \sin(12\pi) - \sin(4\pi) = 0 - 0 = 0. \end{align*} \][/tex]
The calculated values are:
[tex]\[ \begin{array}{c|c} x & H(x) \\ \hline 0 & 0 \\ \frac{\pi}{6} & -\frac{\sqrt{3}}{2} \\ \frac{\pi}{3} & -\frac{\sqrt{3}}{2} \\ \frac{\pi}{2} & 0 \\ \frac{2\pi}{3} & \frac{\sqrt{3}}{2} \\ \frac{5\pi}{6} & \frac{\sqrt{3}}{2} \\ \pi & 0 \\ \frac{7\pi}{6} & -\frac{\sqrt{3}}{2} \\ \frac{4\pi}{3} & -\frac{\sqrt{3}}{2} \\ \frac{3\pi}{2} & 0 \\ \frac{5\pi}{3} & \frac{\sqrt{3}}{2} \\ \frac{11\pi}{6} & \frac{\sqrt{3}}{2} \\ 2\pi & 0 \\ \end{array} \][/tex]
These points can be plotted to create the graph of [tex]\( H(x) = \sin(6x) - \sin(2x) \)[/tex] over the interval [tex]\([0, 2\pi]\)[/tex].
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