Use the Law of Cosines to solve the triangle. Round your answers to two decimal places.
A = 45°,  b = 4,  c = 16

To solve the triangle using the Law of Cosines, we can find the missing side \( a \) and the other two angles \( B \) and \( C \).

### Step 1: Use the Law of Cosines to find side \( a \)

The Law of Cosines states:

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

We are given:
- \( A = 45^\circ \)
- \( b = 4 \)
- \( c = 16 \)

Substitute these values into the equation:

\[
a^2 = 4^2 + 16^2 - 2(4)(16) \cdot \cos(45^\circ)
\]
\[
a^2 = 16 + 256 - 128 \cdot \cos(45^\circ)
\]
Since \( \cos(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.707 \):

\[
a^2 = 16 + 256 - 128 \cdot 0.707
\]
\[
a^2 \approx 272 - 90.496
\]
\[
a^2 \approx 181.504
\]
\[
a \approx \sqrt{181.504} \approx 13.47
\]

### Step 2: Use the Law of Cosines to find angle \( B \)

Now, we can use the Law of Cosines again to find angle \( B \):

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

Substitute the known values:

\[
\cos(B) = \frac{4^2 + 13.47^2 - 16^2}{2(4)(13.47)}
\]
\[
\cos(B) = \frac{16 + 181.504 - 256}{2(4)(13.47)}
\]
\[
\cos(B) = \frac{-58.496}{107.76}
\]
\[
\cos(B) \approx -0.543
\]

Now, find \( B \) by taking the inverse cosine:

\[
B \approx \cos^{-1}(-0.543) \approx 122.91^\circ
\]

### Step 3: Use the sum of angles in a triangle to find angle \( C \)

The sum of the angles in any triangle is \( 180^\circ \). Therefore:

\[
C = 180^\circ - A - B
\]
\[
C = 180^\circ - 45^\circ - 122.91^\circ
\]
\[
C \approx 12.09^\circ
\]

### Final Answer:

- \( a \approx 13.47 \)
- \( B \approx 122.91^\circ \)
- \( C \approx 12.09^\circ \)

Let me know if you'd like further details or explanations.

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Here are 5 related questions:
1. How does the Law of Cosines compare to the Law of Sines?
2. What is the Law of Cosines formula for each angle in a triangle?
3. Can the Law of Cosines be applied to right triangles?
4. How do you find an angle if only the sides are given?
5. What other methods exist for solving oblique triangles?

**Tip:** The Law of Cosines is especially useful when dealing with triangles that are not right-angled.

Use the Law of Cosines to solve the triangle. Round your answers to two decimal places. A = 45°, b = 4, c = 16

Learn how to solve a triangle using the Law of Cosines. Given A = 45°, b = 4, and c = 16, we calculate the missing side a and angles B and C. This problem covers key trigonometric concepts, including the Law of Cosines. Ideal for students in Grades 10-12.

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