Law of Sines and Law of Cosines: Applications in Triangle Solving

Law of Sines

Definition and Formula

The Law of Sines is a fundamental relationship in trigonometry that relates the lengths of sides of a triangle to the sines of its angles. It is especially useful for solving non-right triangles (oblique triangles).

• Formula:

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

• a, b, c are the lengths of the sides opposite angles A, B, C respectively.

• This law is applicable in the following cases: ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and SSA (Side-Side-Angle).

Example Applications

• Solving for a side: Given two angles and one side, use the Law of Sines to find unknown sides.

• Solving for an angle: Given two sides and a non-included angle, use the Law of Sines to find unknown angles.

Worked Example

Given $\frac{7.31}{\sin 74.4^\circ} = \frac{c}{\sin 25.3^\circ}$, solve for c:

$c = \frac{7.31 \times \sin 25.3^\circ}{\sin 74.4^\circ} = 3.24$

Step-by-Step Problem Solving

1. Use the angle sum property to find the missing angle: $A = 180^\circ - B - C$

2. Apply the Law of Sines to solve for unknown sides:

   • $\frac{b}{\sin B} = \frac{c}{\sin C}$

   • $b = \frac{\sin B \times c}{\sin C}$

3. Apply the Law of Sines to solve for unknown angles:

   • $\sin A = \frac{a \sin C}{c}$

   • $A = \sin^{-1}(\text{value})$

Applications

• Solving triangles in navigation, surveying, and physics problems.

• Finding distances and angles in real-world scenarios, such as determining the distance to a helicopter or the separation between planes.

Law of Cosines

Definition and Formula

The Law of Cosines generalizes the Pythagorean theorem for any triangle, relating the lengths of sides to the cosine of one angle. It is especially useful for solving triangles when two sides and the included angle (SAS) or all three sides (SSS) are known.

• Formulas:

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

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

$b^2 = a^2 + c^2 - 2ac \cos B$

• a, b, c are the lengths of the sides opposite angles A, B, C respectively.

• This law is applicable in the following cases: SAS (Side-Angle-Side) and SSS (Side-Side-Side).

Example Applications

• Solving for a side: Given two sides and the included angle, use the Law of Cosines to find the third side.

• Solving for an angle: Given all three sides, use the Law of Cosines to find an angle.

Worked Example

Given $a = 20.3$, $b = 13.7$, $C = 125.3^\circ$, solve for c:

$c^2 = 20.3^2 + 13.7^2 - 2(20.3)(13.7)\cos 125.3^\circ$ $c^2 = 921.196$ $c = 30.4$

Step-by-Step Problem Solving

1. Use the Law of Cosines to solve for unknown sides or angles.

2. Use the Law of Sines to solve for remaining unknowns if needed.

3. Use the angle sum property to find the third angle: $A = 180^\circ - B - C$

Applications

• Solving triangles in navigation, engineering, and physics problems.

• Finding distances between moving objects, such as planes or submarines.

Problem-Solving Strategies

General Steps for Triangle Solving

1. Identify the given information: Determine which sides and angles are known.

2. Choose the appropriate law: Use the Law of Sines for ASA, AAS, SSA cases; use the Law of Cosines for SAS, SSS cases.

3. Apply the formulas: Substitute known values and solve for unknowns.

4. Check for ambiguous cases: In SSA cases, check if the triangle is possible and if there are multiple solutions.

5. Use supplementary angle theorem: For exterior angles, use $B = 180^\circ - \text{known angle}$.

6. Use the angle sum property: $A + B + C = 180^\circ$ for any triangle.

Tables: Triangle Solving Cases

The following table summarizes the main triangle cases and the recommended solving method:

Case	Given	Method
ASA	Two angles, included side	Law of Sines
AAS	Two angles, non-included side	Law of Sines
SSA	Two sides, non-included angle	Law of Sines (ambiguous case)
SAS	Two sides, included angle	Law of Cosines
SSS	Three sides	Law of Cosines

Real-World Applications

• Navigation: Calculating distances and bearings between locations.

• Surveying: Determining inaccessible distances using angles and sides.

• Physics: Analyzing forces and vectors in non-right triangles.

• Engineering: Designing structures and analyzing components with oblique triangles.

Worked Problem Example

Finding the Distance to a Helicopter

1. Use supplementary angle theorem to find missing angle.

2. Use angle sum property to find the third angle.

3. Apply Law of Sines to solve for unknown side.

$b = \frac{\sin B \times c}{\sin C}$

$b = \frac{\sin 136^\circ \times 1540}{\sin 12^\circ} = 5145.3$

Observer A is 5145 m away from the helicopter.

Additional info:

• Velocity formula used in some problems: $v = \frac{d}{t}$ or $d = vt$

• Angle sum property: $A + B + C = 180^\circ$

• Supplementary angle theorem: $\text{Exterior angle} = 180^\circ - \text{interior angle}$