object makes angle tan^-1(3 root 3/5) with an inclined plane of angle 30 degree. what is the angle with the vertical

To find the angle an object makes with the vertical, given it makes an angle \(\theta\) with an inclined plane of angle 30 degrees, we can follow these steps:

1. First, determine \(\theta\):
   \[
   \theta = \tan^{-1}\left(\frac{3\sqrt{3}}{5}\right)
   \]

2. Calculate the numerical value of \(\theta\):
   \[
   \theta = \tan^{-1}\left(\frac{3\sqrt{3}}{5}\right) \approx 45^\circ
   \]
   Note: For exact calculation, the expression should be evaluated using a calculator or further simplification. However, approximating \(\theta\) to 45 degrees simplifies the problem and is a reasonable approximation here.

3. Given the inclined plane makes an angle of \(30^\circ\) with the horizontal, the angle with the vertical is \(90^\circ - 30^\circ = 60^\circ\).

4. The object’s angle with the vertical can be calculated using the fact that it makes an angle \(\theta\) with the inclined plane. Let \(\phi\) be the angle with the vertical. Since the sum of angles around a point is \(180^\circ\), we have:
   \[
   \phi = 90^\circ - (60^\circ - \theta)
   \]
   Given \(\theta \approx 45^\circ\):
   \[
   \phi = 90^\circ - (60^\circ - 45^\circ) = 90^\circ - 15^\circ = 75^\circ
   \]

Therefore, the object makes an angle of approximately \(75^\circ\) with the vertical.

Would you like more details or have any questions?

Here are some other questions you might want to explore:

1. How do you simplify trigonometric expressions?
2. What are the properties of inverse trigonometric functions?
3. How do you convert between degrees and radians?
4. What are some real-world applications of trigonometry?
5. How do inclined planes affect motion and forces in physics?

**Tip:** Always double-check your angle calculations and ensure you're using the correct units (degrees vs. radians) for accurate results.

Learn how to find the angle an object makes with the vertical when it makes an angle \( \theta \) with an inclined plane of 30 degrees. Detailed step-by-step solution using trigonometric principles and inverse trigonometric functions.

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