Question

Question
Explain how you would solve a triangle given two sides and the included angle.

Answer 1

To solve a triangle given two sides and the included angle, you can use the Law of Cosines. This situation typically arises in problems where you know two sides (let's call them \( a \) and \( b \)) and the included angle (let's call it \( C \)). Here’s a step-by-step process to solve for the remaining parts of the triangle (the third side and the other two angles).

### Step 1: Determine the Length of the Third Side
Using the Law of Cosines, you can find the length of the third side \( c \) using the formula:

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

1. Substitute the known values of sides \( a \), \( b \), and angle \( C \) into the formula.
2. Calculate \( c \) by taking the square root:

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

### Step 2: Use the Law of Sines to Find the Other Angles
Now that you have all three sides of the triangle (sides \( a \), \( b \), and \( c \)), you can find the other two angles (\( A \) and \( B \)) using the Law of Sines:

\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\]

1. To find angle \( A \):
\[
\frac{a}{\sin(A)} = \frac{c}{\sin(C)}
\]
Rearranging gives:
\[
\sin(A) = \frac{a \cdot \sin(C)}{c}
\]
Compute \( \sin(A) \) and then angle \( A \) by taking the arcsine function:

\[
A = \arcsin\left(\frac{a \cdot \sin(C)}{c}\right)
\]

2. To find angle \( B \):
Use the sum of angles in a triangle (which is 180 degrees or \( \pi \) radians):
\[
B = 180^\circ - A - C
\]
or in radians:
\[
B = \pi - A - C
\]

### Summary
1. Use the Law of Cosines to find the third side \( c \).
2. Use the Law of Sines to find angle \( A \).
3. Calculate angle \( B \) using the triangle angle sum.

With these steps, you can fully solve the triangle given two sides and the included angle.