Question

Use the inverse of cosine to find the approximate interior angle measure at vertex A . Round your answer to the nearest whole degree. (1 point)

The length of the side A and B is 17, the length of the side B and C is 15, the length of the side between A and C is 8

Answer 1

To find the interior angle at vertex A using the lengths of the sides, we can apply the Law of Cosines. The Law of Cosines states:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

In this case, we can denote the sides as follows:

- Let side a (the length of line segment BC) = 15
- Let side b (the length of line segment AC) = 8
- Let side c (the length of line segment AB) = 17

We are interested in finding angle A (the angle opposite side a). According to the Law of Cosines, we can rewrite the formula to solve for \(\cos(A)\):

\[
a^2 = b^2 + c^2 - 2bc \cdot \cos(A)
\]

Substituting in our values, we have:

\[
15^2 = 8^2 + 17^2 - 2 \cdot 8 \cdot 17 \cdot \cos(A)
\]

Calculating the squares:

\[
225 = 64 + 289 - 272 \cdot \cos(A)
\]

Now combine the constants:

\[
225 = 353 - 272 \cdot \cos(A)
\]

Isolating \(\cos(A)\):

\[
272 \cdot \cos(A) = 353 - 225
\]
\[
272 \cdot \cos(A) = 128
\]
\[
\cos(A) = \frac{128}{272}
\]

Simplifying \(\frac{128}{272}\):

\[
\cos(A) = \frac{16}{34} = \frac{8}{17}
\]

Now, we use the inverse cosine function to find the angle A:

\[
A = \cos^{-1}\left(\frac{8}{17}\right)
\]

Calculating using a calculator (ensuring it's in degrees):

\[
A \approx 57.12^\circ
\]

Rounding to the nearest whole degree:

\[
A \approx 57^\circ
\]

Thus, the approximate interior angle measure at vertex A is **57 degrees**.