Question

Use the inverse of cosine to find the approximate interior angle measure at vertex A. Round your answer to the nearest whole degree.
Lengths a to c 8
A to b 17 and
C to b 15

Answer 1

To find the approximate interior angle measure at vertex A using the cosine rule, we can use the formula:

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

where:
- \( a = 8 \) (length between points B and C),
- \( b = 17 \) (length between points A and B),
- \( c = 15 \) (length between points A and C).

Substituting in the values:

1. Calculate \( b^2 \), \( c^2 \), and \( a^2 \):
\[
b^2 = 17^2 = 289
\]
\[
c^2 = 15^2 = 225
\]
\[
a^2 = 8^2 = 64
\]

2. Substitute the values into the cosine formula:
\[
\cos(A) = \frac{289 + 225 - 64}{2 \cdot 17 \cdot 15}
\]

3. Simplify the numerator:
\[
289 + 225 - 64 = 450
\]

4. Calculate the denominator:
\[
2 \cdot 17 \cdot 15 = 510
\]

5. Substitute back into the equation:
\[
\cos(A) = \frac{450}{510}
\]

6. Simplify the fraction:
\[
\cos(A) = \frac{15}{17} \approx 0.8824
\]

7. Now, take the inverse cosine to find angle A:
\[
A \approx \cos^{-1}(0.8824)
\]

Using a calculator to find \( A \):
\[
A \approx 28.96^\circ
\]

8. Rounding to the nearest whole degree:
\[
A \approx 29^\circ
\]

Thus, the approximate interior angle measure at vertex A is **29 degrees**.