Question

Use the law of cosines to find value of angle A in a right triangle. a=9 b=40 c=41

Answer 1

In a right triangle, the Law of Cosines can be simplified by using the fact that one of the angles is \(90^\circ\). However, we need to find angle \(A\) using the formula for the Law of Cosines, which is:

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

Here:
- \(c\) is the side opposite angle \(A\) (the hypotenuse in a right triangle).
- \(a\) and \(b\) are the other two sides.

Given:
- \(a = 9\)
- \(b = 40\)
- \(c = 41\)

Now, we can plug in the values:

\[
41^2 = 9^2 + 40^2 - 2 \cdot 9 \cdot 40 \cdot \cos(A)
\]

Calculating each term:

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

Now substituting these values into the equation:

\[
1681 = 81 + 1600 - 2 \cdot 9 \cdot 40 \cdot \cos(A)
\]

Now simplify:

\[
1681 = 1681 - 720 \cdot \cos(A)
\]

Next, we can rearrange this equation to solve for \(\cos(A)\):

\[
0 = -720 \cdot \cos(A)
\]

Thus:

\[
\cos(A) = 0
\]

The cosine of angle \(A\) being zero indicates that angle \(A\) is \(90^\circ\).

So, angle \(A\) is:

\[
\boxed{90^\circ}
\]