Question

If sine of x degrees equals four-fifths, what is the value of b?

triangle LMN in which angle M measures 90 degrees, angle L measures x degrees, LN measures 22.5 units, and NM measures 3b units

b = 4
b = 5
b = 6
b = 7

Answer 1

In triangle LMN, we know the following:

- Angle M is a right angle (90 degrees).
- Angle L measures \( x \) degrees.
- The side \( LN \) (the hypotenuse) measures 22.5 units.
- The side \( NM \) (an opposite side to angle L) measures \( 3b \) units.

Given that \( \sin(x) = \frac{4}{5} \), we can use this information to set up the sine definition for angle L:

\[
\sin(L) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{NM}{LN}
\]

Substituting the lengths we have:

\[
\sin(x) = \frac{3b}{22.5}
\]

Now, plugging in the value of \( \sin(x) \):

\[
\frac{4}{5} = \frac{3b}{22.5}
\]

To solve for \( b \), we can cross-multiply:

\[
4 \cdot 22.5 = 3b \cdot 5
\]
\[
90 = 15b
\]

Now, isolate \( b \):

\[
b = \frac{90}{15} = 6
\]

Thus, the value of \( b \) is:

\[
\boxed{6}
\]