Question

Use the Law of Cosines, a2=b2+c2−2bc⋅cosA , to find the value of angle A in a right triangle, where a=9 , b=40 , and c=41

Answer 1

To find angle \( A \) in the triangle using the Law of Cosines, we can rearrange the formula:

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

We need to isolate \( \cos A \):

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

Now we can plug in the values \( a = 9 \), \( b = 40 \), and \( c = 41 \):

1. Calculate \( b^2 \), \( c^2 \), and \( a^2 \):

\[
b^2 = 40^2 = 1600
\]
\[
c^2 = 41^2 = 1681
\]
\[
a^2 = 9^2 = 81
\]

2. Substitute these values into the formula for \( \cos A \):

\[
\cos A = \frac{1600 + 1681 - 81}{2 \cdot 40 \cdot 41}
\]

3. Simplify the numerator:

\[
1600 + 1681 - 81 = 3200
\]

4. Calculate the denominator:

\[
2 \cdot 40 \cdot 41 = 3280
\]

5. Now substitute into the \( \cos A \):

\[
\cos A = \frac{3200}{3280}
\]

6. Simplify:

\[
\cos A = \frac{320}{328} = \frac{80}{82} = \frac{40}{41}
\]

7. Now, to find angle \( A \), take the arccosine:

\[
A = \cos^{-1}\left(\frac{40}{41}\right)
\]

8. Using a calculator to find \( A \):

\[
A \approx \cos^{-1}\left(0.97561\right) \approx 12.53^\circ
\]

Thus, the value of angle \( A \) is approximately \( 12.53^\circ \).