Answer:
Certainly! To solve using the sine law, given that the angle at the top of the tree is 20 degrees and the tree is 18 feet 30 ft away, we need to find either the height of the tree or the distance from the tree to the point where the angle is measured.
Let's assume that \(A\) is the angle at the top of the tree (20 degrees), \(a\) is the height of the tree (opposite \(A\)), \(B\) is the angle at the base (18 degrees), and \(b\) is the distance from the tree to the point where the angle is measured (opposite \(B\)).
Using the sine law, we have:
\[\frac{a}{\sin A} = \frac{b}{\sin B}\]
Plug in the given values:
\[\frac{a}{\sin 20^\circ} = \frac{18.5}{\sin 18^\circ}\]
Now, solve for \(a\):
\[a = \frac{18.5 \cdot \sin 20^\circ}{\sin 18^\circ}\]
Calculate the value of \(a\) using a calculator. This will give you the height of the tree.
Keep in mind that the angles and sides should be in the same units (either degrees or radians) throughout the calculation. Also, ensure that your calculator is set to the appropriate unit mode (degrees or radians) when using trigonometric functions.