Study Notes: Law of Sines and Law of Cosines for Oblique Triangles

Oblique Triangles

Definition and Classification

An oblique triangle is a triangle that does not contain a right angle. Oblique triangles are classified as follows:

• Acute triangle: All angles are less than 90°.

• Obtuse triangle: One angle is greater than 90°.

To solve oblique triangles (i.e., to find unknown sides or angles), the Law of Sines and Law of Cosines are used. The primary trigonometric functions and the Pythagorean Theorem are not applicable for non-right triangles.

Law of Sines

Statement of the Law

The Law of Sines relates the sides and angles of an oblique triangle as follows:

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

where A, B, and C are the angles, and a, b, and c are the sides opposite those angles, respectively.

Cases for Applying the Law of Sines

• Case 1 (SAA or ASA): Given two angles and one side.

• Case 2 (SSA): Given two sides and an angle opposite one of them (the "Ambiguous Case").

Case 1: SAA or ASA (Side-Angle-Angle or Angle-Side-Angle)

• Find the third angle using the angle sum property: $A + B + C = 180^\circ$.

• Use the Law of Sines to find the unknown sides.

Example: Given $A = 106.5^\circ$, $B = 23.7^\circ$, $a = 51$ cm.

• Find $C$: $C = 180^\circ - 106.5^\circ - 23.7^\circ = 49.8^\circ$

• Find $b$: $\displaystyle \frac{\sin 23.7^\circ}{b} = \frac{\sin 106.5^\circ}{51}$, so $b = \frac{51 \sin 23.7^\circ}{\sin 106.5^\circ} = 21.4$ cm

• Find $c$: $\displaystyle \frac{\sin 49.8^\circ}{c} = \frac{\sin 106.5^\circ}{51}$, so $c = \frac{51 \sin 49.8^\circ}{\sin 106.5^\circ} = 40.6$ cm

Case 2: SSA (Side-Side-Angle) – The Ambiguous Case

• Use the Law of Sines to find the unknown angle.

• Use the angle sum to find the third angle.

• Use the Law of Sines to find the remaining side.

• Ambiguity: There may be one, two, or no solutions depending on the given values.

Example: Given $B = 25.1^\circ$, $c = 77.3$ cm, $b = 41.4$ cm.

• Find $C$ using Law of Sines: $\displaystyle \frac{\sin C}{c} = \frac{\sin B}{b}$

• First solution: $C = 52.4^\circ$, $A = 180^\circ - 25.1^\circ - 52.4^\circ = 102.5^\circ$, $a = 95.3$ cm

• Second solution: $C' = 180^\circ - 52.4^\circ = 127.6^\circ$, $A' = 180^\circ - 25.1^\circ - 127.6^\circ = 27.3^\circ$, $a' = 44.8$ cm

Note: If the side opposite the given angle is shorter than the other given side, two solutions are possible. If it is longer, only one solution exists.

Applications of the Law of Sines

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

Example: Two observers A and B are 1540 m apart and sight a helicopter due east at angles of elevation 32.0° and 44.0°. Find the distance from observer A to the helicopter.

Solution: Use the Law of Sines to set up the triangle and solve for the unknown distance. (Answer: 5145 m)

Law of Cosines

Statement of the Law

The Law of Cosines generalizes the Pythagorean Theorem for any triangle:

• $c^2 = a^2 + b^2 - 2ab \cos C$

• $b^2 = a^2 + c^2 - 2ac \cos B$

• $a^2 = b^2 + c^2 - 2bc \cos A$

If the included angle is 90°, $\cos 90^\circ = 0$, and the formula reduces to the Pythagorean Theorem.

Cases for Applying the Law of Cosines

• Case 3 (SAS): Given two sides and the included angle.

• Case 4 (SSS): Given all three sides.

Case 3: SAS (Side-Angle-Side)

• Use the Law of Cosines to find the third side.

• Use the Law of Sines to find the smallest angle.

• Use the angle sum to find the remaining angle.

Example: Given $A = 62.1^\circ$, $b = 30.0$ cm, $c = 75.0$ cm.

• Find $a$ using Law of Cosines: $a^2 = b^2 + c^2 - 2bc \cos A$

• Find $B$ using Law of Sines: $\displaystyle \frac{\sin B}{b} = \frac{\sin A}{a}$

• Find $C$ using angle sum: $C = 180^\circ - A - B$

Case 4: SSS (Side-Side-Side)

• Use the Law of Cosines to find the largest angle (opposite the longest side).

• Use the Law of Sines to find another angle.

• Use the angle sum to find the third angle.

Example: Given $a = 49.33$ cm, $b = 21.61$ cm, $c = 42.57$ cm.

• Find $A$ using Law of Cosines: $a^2 = b^2 + c^2 - 2bc \cos A$

• Find $B$ using Law of Sines: $\displaystyle \frac{\sin B}{b} = \frac{\sin A}{a}$

• Find $C$ using angle sum: $C = 180^\circ - A - B$

Applications of the Law of Cosines

• Solving triangles in navigation, engineering, and physics.

Example: A plane leaves an airport and travels 624 km due east, then turns north and travels 326 km, then turns again and travels 846 km back to the airport. Find the angles of the turns.

Solution: Use the Law of Cosines to solve for the unknown angles. (First turn: 57.3°, second turn: 141.7°)

Summary Table: Laws for Solving Oblique Triangles

Case	Given	Law to Use	Notes
1 (SAA/ASA)	2 angles, 1 side	Law of Sines	Find third angle, then other sides
2 (SSA)	2 sides, angle opposite one	Law of Sines	Ambiguous case: 0, 1, or 2 solutions
3 (SAS)	2 sides, included angle	Law of Cosines	Find third side, then angles
4 (SSS)	3 sides	Law of Cosines	Find largest angle first

Key Formulas

• Law of Sines: $\displaystyle \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

• Law of Cosines:

  • $c^2 = a^2 + b^2 - 2ab \cos C$

  • $b^2 = a^2 + c^2 - 2ac \cos B$

  • $a^2 = b^2 + c^2 - 2bc \cos A$

Practice Problems

• Solve the triangle with $B = 121^\circ$, $C = 44.2^\circ$, $c = 729$ cm.

• Two planes are due east from a control tower at distances 15.8 km and 32.7 km, with angles of elevation 26.4° and 12.4°. How far apart are the planes?

• A submarine travels 23.5 km/h for 2 hours at 32.1° north of west, then turns 21.5° more north of west for 1 hour. How far is it from its base?

Answers and detailed solutions can be found in the exercises above.

Additional info: The ambiguous case (SSA) is a classic scenario in trigonometry where the Law of Sines may yield two possible triangles. Always check for the possibility of two solutions by considering the sine inverse and its supplement.