Type the correct answer in the box.

What trigonometric expression can be used to find the value of [tex]\( x \)[/tex]? Replace [tex]\( a \)[/tex] and [tex]\( b \)[/tex] with the correct values.

[tex]\[
\frac{a}{\tan (b)}
\][/tex]



Answer :

To find the value of [tex]\( x \)[/tex] using the given trigonometric expression for a right triangle where [tex]\( a \)[/tex] is the length of the side adjacent to the angle [tex]\( b \)[/tex]:

1. The trigonometric expression we need to use is:
[tex]\[ x = \frac{a}{\tan(b)} \][/tex]

2. We are given specific values for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- [tex]\( a = 5 \)[/tex] (the length of the adjacent side)
- [tex]\( b = 30^\circ \)[/tex] (the measure of the angle in degrees)

3. Substitute these values into the trigonometric expression:
[tex]\[ x = \frac{5}{\tan(30^\circ)} \][/tex]

4. Recall that [tex]\( \tan(30^\circ) \)[/tex] is a known value in trigonometry. Specifically:
[tex]\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.57735 \][/tex]

5. Therefore, substituting this into the equation gives us:
[tex]\[ x = \frac{5}{0.57735} \][/tex]

6. Performing the division:
[tex]\[ x \approx 8.660254037844387 \][/tex]

Thus, the value of [tex]\( x \)[/tex] using the given trigonometric expression with the provided values is approximately [tex]\( 8.660254037844387 \)[/tex].