Let's solve the given trigonometric equation step-by-step and find all solutions for \(x\) in the interval \([0, 2\pi)\). The given equation is:
[tex]\[ 5 \sin(2x) = -7.5 \cos(x) \][/tex]
To solve this equation, follow these steps:
### Step 1: Express \(\sin(2x)\) in terms of \(\sin(x)\) and \(\cos(x)\)
Recall the double-angle identity:
[tex]\[ \sin(2x) = 2 \sin(x) \cos(x) \][/tex]
Using this identity, rewrite the equation:
[tex]\[ 5 \cdot 2 \sin(x) \cos(x) = -7.5 \cos(x) \][/tex]
[tex]\[ 10 \sin(x) \cos(x) = -7.5 \cos(x) \][/tex]
### Step 2: Simplify the equation
Since \(\cos(x) \neq 0\) at some points in the interval \([0, 2\pi)\), we can divide both sides by \(\cos(x)\) (so long as \(\cos(x) \neq 0\)):
[tex]\[ 10 \sin(x) = -7.5 \][/tex]
[tex]\[ \sin(x) = -0.75 \][/tex]
### Step 3: Find the general solutions for \(\sin(x) = -0.75\)
We use the inverse sine function to find the solutions for \(x\):
[tex]\[ x = \sin^{-1}(-0.75) \][/tex]
The principal value of \( x \) in the interval \([-\pi/2, \pi/2]\) is:
[tex]\[ x = -\sin^{-1}(0.75) \approx -0.8481 \][/tex]
Since this lies outside the interval \([0, 2\pi)\), we need to adjust the angle to lie within the given interval. Recall sine's symmetry properties to find appropriate solutions. Specifically, the sine function gives solutions in the form:
[tex]\[ x = \pi + \sin^{-1}(0.75) \quad \text{and} \quad x = 2\pi - \sin^{-1}(0.75) \][/tex]
Where we account for symmetry about \(\pi\) and \(2\pi -\theta\):
[tex]\[ x_1 = \pi + \sin^{-1}(0.75) \][/tex]
[tex]\[
x_2 = 2\pi - \sin^{-1}(0.75)
\approx 3.9900, 5.4351 \text{(since \text{2}\pi}\approx6.2832 \)
### Step 4: Collect additional solutions
Since \(\cos(x) = 0\) also provides some critical angles and additional points like \(\pi/2\) and \(3\pi/2\), re-estimated these angles would be:
\[ \boxed{x = \frac{\pi}{2}} \][/tex]
[tex]\[ x_4 =3/2 \pi \][/tex]
### Step 5: Arrange the solutions in the interval \([0, 2\pi)\)
Summarizing the steps above, the solutions to the equation in the interval \([0, 2\pi)\) are:
[tex]\[ x_1 \approx 1.5708\][/tex]
[tex]\[ x_2 \approx 3.9900\][/tex]
[tex]\[ x \approx 4.7124\][/tex]
[tex]\[ x_3 \approx5.4351\][/tex]
Arranging solutions in increasing order:
[tex]\[ x \approx 1.5708,3.9900, 4.7124, 5.4351 \][/tex]
These are the desired approximate solutions to four decimal places.
So the solutions to the equation \(5 \sin(2x) = -7.5 \cos(x)\) in the interval \([0, 2\pi)\) are:
[tex]\[
\boxed{
\begin{array}{l}
x \approx 1.5708 \text{ (smallest value) } \\
x \approx 3.9900 \\
x \approx 4.7124 \\
x \approx 5.4351 \text{ (largest value) }
\end{array}
}
\][/tex]
