Question

Use the tangent ratio to solve for missing side length AB of the right triangle. Round to the nearest hundredths (2 decimal places).
line BC= 15
angle C= 15 degrees

Answer 1

To solve for the length of side AB in the right triangle using the tangent ratio, we can use the following relationship:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

In this triangle:
- Angle \(C\) is 15 degrees.
- Side \(BC\) (the side opposite angle \(C\)) is 15.
- Side \(AB\) (the side adjacent to angle \(C\)), which we want to find.

Using the tangent ratio for angle \(C\):

\[
\tan(15^\circ) = \frac{BC}{AB}
\]

Substituting the known values:

\[
\tan(15^\circ) = \frac{15}{AB}
\]

Now, we can solve for \(AB\):

1. Calculate \(\tan(15^\circ)\):
- Using a calculator, \(\tan(15^\circ) \approx 0.2679\).

2. Substitute this value back into the equation:

\[
0.2679 = \frac{15}{AB}
\]

3. Rearranging to find \(AB\):

\[
AB \cdot 0.2679 = 15
\]

\[
AB = \frac{15}{0.2679}
\]

4. Calculating \(AB\):

\[
AB \approx \frac{15}{0.2679} \approx 55.94
\]

Therefore, the length of side \(AB\) rounded to the nearest hundredths is:

\[
\boxed{55.94}
\]