Question

What are the exact values of a and b?

Right triangle A B C is shown with labeled sides and one labeled angle. Side A B is the hypotenuse and it is labeled 7. Side B C is the height and it is labeled a. Side C A is the base and it is labeled b. The angle opposite side B C is 30 degrees.

Answer 1

In triangle ABC, where AB is the hypotenuse measuring 7 units, BC is the height (labeled as \( a \)), CA is the base (labeled as \( b \)), and the angle opposite side BC (angle A) is 30 degrees, we can use trigonometric ratios to find the values for \( a \) and \( b \).

### To find \( a \):

Using the sine function, which relates the opposite side to the hypotenuse in a right triangle:

\[
\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AB}
\]

Substituting the known values:

\[
\sin(30^\circ) = \frac{a}{7}
\]

Since \( \sin(30^\circ) = \frac{1}{2} \), we can write:

\[
\frac{1}{2} = \frac{a}{7}
\]

Now, solving for \( a \):

\[
a = 7 \times \frac{1}{2} = 3.5
\]

### To find \( b \):

Using the cosine function, which relates the adjacent side to the hypotenuse:

\[
\cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{CA}{AB}
\]

Substituting the known values:

\[
\cos(30^\circ) = \frac{b}{7}
\]

Knowing \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), we can write:

\[
\frac{\sqrt{3}}{2} = \frac{b}{7}
\]

Now, solving for \( b \):

\[
b = 7 \times \frac{\sqrt{3}}{2} = \frac{7\sqrt{3}}{2}
\]

### Final Values

Thus, the exact values are:

\[
a = 3.5
\]

\[
b = \frac{7\sqrt{3}}{2}
\]