Question

Use the inverse of the tangent ratio to find the approximate angle measure at vertex A
𝐴 . Round your answer to the nearest whole degree.

b 20
c= 25
a ?

Answer 1

ts aint right

Answer 2

To find the angle measure at vertex A using the inverse tangent ratio, we first need to identify the correct sides of the triangle corresponding to the angle A.

In a triangle, if we denote:
- \( a \) as the length of the side opposite angle A,
- \( b \) as the length of the side adjacent to angle A,
- \( c \) as the length of the hypotenuse,

the tangent of angle A can be given by the formula:

\[
\tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b}
\]

Since we need to find \( A \), the formula can be rearranged using the inverse tangent (also known as arctan):

\[
A = \tan^{-1}\left(\frac{a}{b}\right)
\]

In your case, you have:
- \( b = 20 \)
- \( c = 25 \)

We need to find side \( a \) first. Since we know \( b \) and \( c \), we can find \( a \) using the Pythagorean theorem:

\[
c^2 = a^2 + b^2
\]

Substituting the values:

\[
25^2 = a^2 + 20^2
\]
\[
625 = a^2 + 400
\]
\[
a^2 = 625 - 400
\]
\[
a^2 = 225
\]
\[
a = \sqrt{225} = 15
\]

Now we can find angle \( A \):

\[
A = \tan^{-1}\left(\frac{a}{b}\right) = \tan^{-1}\left(\frac{15}{20}\right) = \tan^{-1}(0.75)
\]

Using a calculator to find \( \tan^{-1}(0.75) \):

\[
A \approx 36.87^\circ
\]

Rounding this to the nearest whole degree, we get:

\[
A \approx 37^\circ
\]

Thus, the approximate angle measure at vertex \( A \) is **37 degrees**.