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 10. 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 labeled as \( 10 \), \( BC \) is the height labeled as \( a \), and \( CA \) is the base labeled as \( b \), we know that the angle opposite side \( BC \) (height) is \( 30^\circ \).

Since triangle \( ABC \) is a right triangle, we can use the relationship of the sides in a right triangle along with trigonometric functions.

1. **Using the sine function to find \( a \)**:
\[
\sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{10}
\]
The sine of \( 30^\circ \) is \( \frac{1}{2} \):
\[
\frac{1}{2} = \frac{a}{10}
\]
To find \( a \), we can multiply both sides by \( 10 \):
\[
a = 10 \cdot \frac{1}{2} = 5
\]

2. **Using the cosine function to find \( b \)**:
\[
\cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{10}
\]
The cosine of \( 30^\circ \) is \( \frac{\sqrt{3}}{2} \):
\[
\frac{\sqrt{3}}{2} = \frac{b}{10}
\]
To find \( b \), we multiply both sides by \( 10 \):
\[
b = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}
\]

Thus, the exact values are:
\[
a = 5, \quad b = 5\sqrt{3}.
\]