We can use the definitions of sine, cosine, and tangent to find angle \( a \) in your triangle. Given the information:
- \( ab = 12 \) (adjacent to angle \( a \))
- \( bc = 5 \) (opposite to angle \( a \))
- \( ac = 13 \) (hypotenuse)
Using the tangent function:
The tangent of angle \( a \) is defined as the ratio of the opposite side to the adjacent side:
\[ \tan(a) = \frac{\text{opposite}}{\text{adjacent}} = \frac{bc}{ab} = \frac{5}{12} \]
Finding angle \( a \):
To find angle \( a \), we take the arctangent (inverse tangent) of \( \frac{5}{12} \):
\[ a = \tan^{-1}\left(\frac{5}{12}\right) \]
Calculating this using a calculator:
\[ a \approx 22.62^\circ \]
Thus, the measure of angle \( a \) is approximately \( 22.62^\circ \).