To find the angle at vertex B (angle B) using the inverse tangent ratio, we need to first clarify the setup of the triangle based on your description.
We have:
- \( b \) at the top (angle B)
- \( c \) is a right angle (angle C = 90 degrees)
- \( a \) is the angle at the bottom (angle A)
You mentioned the lengths \( bc = 20 \) (the length of the side opposite angle A) and \( ca = 25 \) (the length of the side adjacent to angle A).
To find angle B (opposite side \( ac \)), we can employ the tangent function:
\[ \tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{bc}{ca} \]
First, let's substitute the values:
\[ \tan(B) = \frac{20}{25} \]
Now, calculate \(\tan(B)\):
\[ \tan(B) = \frac{20}{25} = 0.8 \]
Next, to find angle B, we use the inverse tangent (arctan) function:
\[ B = \tan^{-1}(0.8) \]
Using a calculator:
\[ B \approx 38.66 \text{ degrees} \]
Rounding to the nearest whole degree, we find:
\[ B \approx 39 \text{ degrees} \]
Thus, the approximate angle measure at vertex B is 39 degrees.