Find all angles, $0^{\circ} \leq x< 360^{\circ}$, that satisfy the equation below, to the nearest 1oth of a degree.
\[
5 \sin x \cos x=7 \cos x
\]

Step 1 ：The given equation is a trigonometric equation. To solve it, we can first isolate the variable x. We can divide both sides of the equation by \( \cos x \) (assuming \( \cos x \neq 0 \)) to get \( 5 \sin x = 7 \). Then we can solve for x by taking the inverse sine of both sides. However, we need to consider the range of x and the periodicity of the sine function to find all solutions. We also need to consider the case where \( \cos x = 0 \), which would make the original equation undefined.

Step 2 ：The code has returned six angles that satisfy the equation: 89.8, 89.9, 90.0, 90.1, 90.2, and 270.0 degrees. However, we need to check if these angles are valid solutions. The angles 89.8, 89.9, 90.0, 90.1, 90.2 are close to 90 degrees, where \( \cos x \) is close to 0. This could potentially make the original equation undefined, so we need to verify these solutions. The angle 270 degrees is where \( \cos x \) is exactly 0, which would make the original equation undefined, so this is not a valid solution.

Step 3 ：The code has returned an empty list, which means none of the angles found previously are valid solutions. This is because the cosine of these angles is too close to 0, which would make the original equation undefined. Therefore, there are no angles in the range \( 0^{\circ} \leq x<360^{\circ} \) that satisfy the equation \( 5 \sin x \cos x=7 \cos x \).

Step 4 ：Final Answer: There are no solutions, so the answer is \( \boxed{\text{No Solution}} \).