Question

How do I solve an oblique triangle?

Answer 1

To solve an oblique triangle (a triangle that does not have a right angle), you can use the following methods depending on the information you have:

### 1. **Using the Law of Sines**
The Law of Sines relates the sides and angles of a triangle. It is applicable when you know:

- **Two angles and one side (AAS or ASA)**
- **Two sides and a non-included angle (SSA)**

The Law of Sines states:

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

Where:
- \(a\), \(b\), and \(c\) are the lengths of the sides opposite angles \(A\), \(B\), and \(C\) respectively.

**Steps:**
1. Start by finding any missing angles if you have two angles (AAS or ASA).
2. Use the Law of Sines to solve for the unknown sides or angles.
3. For SSA cases, be mindful that there may be two possible solutions (the ambiguous case).

### 2. **Using the Law of Cosines**
The Law of Cosines is useful when you know:

- **Two sides and the included angle (SAS)**
- **All three sides (SSS)**

The Law of Cosines states:

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

**Steps:**
1. For SAS cases, you can first find the third side using the Law of Cosines, then use the Law of Sines to find the other angles.
2. For SSS cases, use the Law of Cosines to find one angle first, and then apply the Law of Sines to find the other angles.

### Example Problem:
Suppose you know two sides and the included angle (SAS):
- \(a = 7\)
- \(b = 9\)
- \(C = 60^\circ\)

**1. Use the Law of Cosines to find \(c\):**
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
\[
c^2 = 7^2 + 9^2 - 2 \cdot 7 \cdot 9 \cdot \cos(60^\circ)
\]
\[
= 49 + 81 - 63
\]
\[
= 67 \implies c = \sqrt{67}
\]

**2. Use the Law of Sines to find the remaining angles:**
\[
\frac{7}{\sin A} = \frac{9}{\sin B} = \frac{\sqrt{67}}{\sin(60^\circ)}
\]

Solve for \(A\) and \(B\), using the relationship between the angles knowing that \(A + B + C = 180^\circ\).

### Conclusion
Keep this structured approach in mind when solving oblique triangles, and choose the method that suits the provided information best.