evaluate it exactly and give answer in fraction please

To evaluate the given angle exactly and provide the answer in fractions, let's proceed with the exact trigonometric calculations.

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

2. Let's find the exact value of \(\theta\):
   We need to find an angle \(\theta\) such that:
   \[
   \tan(\theta) = \frac{3\sqrt{3}}{5}
   \]

Since \(\tan^{-1}\left(\frac{3\sqrt{3}}{5}\right)\) doesn't correspond to a standard angle, we proceed by keeping it in its arctan form for precision.

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:
   \[
   \phi = 90^\circ - (60^\circ - \theta)
   \]
   Therefore:
   \[
   \phi = 90^\circ - (60^\circ - \tan^{-1}\left(\frac{3\sqrt{3}}{5}\right))
   \]
   Simplifying further:
   \[
   \phi = 90^\circ - 60^\circ + \tan^{-1}\left(\frac{3\sqrt{3}}{5}\right)
   \]
   \[
   \phi = 30^\circ + \tan^{-1}\left(\frac{3\sqrt{3}}{5}\right)
   \]

To convert the angle \(\phi\) to radians and represent it as a fraction:

1. \(30^\circ\) in radians is \(\frac{\pi}{6}\).
2. \(\tan^{-1}\left(\frac{3\sqrt{3}}{5}\right)\) can be left as it is in its exact form.

Thus:
\[
\phi = \frac{\pi}{6} + \tan^{-1}\left(\frac{3\sqrt{3}}{5}\right)
\]

This is the exact angle the object makes with the vertical in radians. If you need this in degrees, it would be:
\[
\phi = 30^\circ + \tan^{-1}\left(\frac{3\sqrt{3}}{5}\right)
\]

Would you like more details or have any questions?

Here are some other questions you might want to explore:

1. How to convert between degrees and radians?
2. What are the properties of inverse trigonometric functions?
3. How do you solve trigonometric equations exactly?
4. What is the significance of angles in trigonometry?
5. How do inclined planes affect motion and forces in physics?

**Tip:** When dealing with trigonometric functions, keeping expressions in their exact forms often helps maintain precision, especially in complex calculations.

Learn how to evaluate an angle exactly and provide the answer in fractions using the inverse tangent function. This problem involves trigonometric calculations and the precise determination of angles, essential for high school students studying trigonometry.

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