Failure is not an option. $$ \begin{array}{l} ext { 10. Solve for the angles of the triangle described below. Express all angles in degrees and round } \ ext { to the nearest hundredth. } \ a=8, b=5, c=4\end{array} $$ (12 pts.)

Question

Failure is not an option. $$ \begin{array}{l}	ext { 10. Solve for the angles of the triangle described below. Express all angles in degrees and round } \ 	ext { to the nearest hundredth. } \ a=8, b=5, c=4\end{array} $$ (12 pts.)

UpStudy AI · Accepted Answer

To solve for the angles of the triangle using the lengths

,

, and

, we can apply the Law of Cosines. First, we’ll calculate angle

using the formula

. Plugging in the values, we get:

Next, let’s calculate angle

using

:

Finally, angle

can be found as

:

Thus, the angles of the triangle are approximately

,

, and

.

$$
c^2 = a^2 + b^2 - 2ab \cos(C)
$$

We can rearrange this formula to solve for angle $$ C $$:

$$
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
$$

### Step 1: Calculate angle $$ C $$

Substituting the values of $$ a $$, $$ b $$, and $$ c $$:

$$
\cos(C) = \frac{8^2 + 5^2 - 4^2}{2 \cdot 8 \cdot 5}
$$

Calculating the squares:

$$
\cos(C) = \frac{64 + 25 - 16}{80}
$$
$$
\cos(C) = \frac{73}{80}
$$

Now, calculate $$ C $$:

$$
C = \cos^{-1}\left(\frac{73}{80}\right)
$$

Using a calculator:

$$
C \approx 43.57^\circ
$$

### Step 2: Calculate angle $$ A $$

Next, we can use the Law of Cosines again to find angle $$ A $$:

$$
a^2 = b^2 + c^2 - 2bc \cos(A)
$$

Rearranging gives:

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

Substituting the values:

$$
\cos(A) = \frac{5^2 + 4^2 - 8^2}{2 \cdot 5 \cdot 4}
$$

Calculating the squares:

$$
\cos(A) = \frac{25 + 16 - 64}{40}
$$
$$
\cos(A) = \frac{-23}{40}
$$

Now, calculate $$ A $$:

$$
A = \cos^{-1}\left(\frac{-23}{40}\right)
$$

Using a calculator:

$$
A \approx 128.37^\circ
$$

### Step 3: Calculate angle $$ B $$

Finally, we can find angle $$ B $$ using the fact that the sum of angles in a triangle is $$ 180^\circ $$:

$$
B = 180^\circ - A - C
$$

Substituting the values we found:

$$
B = 180^\circ - 128.37^\circ - 43.57^\circ
$$
$$
B \approx 8.06^\circ
$$

### Summary of the angles

Thus, the angles of the triangle are approximately:

- $$ A \approx 128.37^\circ $$
- $$ B \approx 8.06^\circ $$
- $$ C \approx 43.57^\circ $$

### Final Answer

$$
\begin{align*}
A & \approx 128.37^\circ \
B & \approx 8.06^\circ \
C & \approx 43.57^\circ \
\end{align*}
$$
The angles of the triangle are approximately $$ A \approx 128.37^\circ $$, $$ B \approx 8.06^\circ $$, and $$ C \approx 43.57^\circ $$.

Failure is not an option.

(12 pts.)

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Failure is not an option. (12 pts.)