Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. Enter your answers from smallest to largest.

[tex]\[
\begin{array}{l}
5 \sin 2x = -7.5 \cos x \quad [0, 2\pi) \\
x = \square \quad \text{(smallest value)} \\
x = \square \quad \text{(largest value)} \\
x = \square \\
x = \square
\end{array}
\][/tex]



Answer :

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]