12.2 Law of Sines and Law of Cosines

The Law of Sines and Law of Cosines are powerful tools for solving triangles without right angles. These laws extend trigonometric concepts beyond right triangles, allowing us to tackle more complex geometric problems in real-world situations.

Understanding these laws is crucial for applications in navigation, surveying, and engineering. They provide methods to find missing sides or angles in triangles, with the Law of Cosines being particularly useful when we have three sides but no angles.

Solving triangles with the Law of Sines

Using the Law of Sines with two angles and a side

• The Law of Sines states that in any triangle, the ratio of the sine of an angle to the length of the opposite side is constant for all three angles and their corresponding opposite sides: $\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$
• When given two angles and any side (SAS or ASA), the third angle can be calculated using the fact that the sum of the angles in a triangle is 180°
  • Example: If angle A is 30°, angle B is 45°, and side c is 10 units long, angle C can be calculated as 180° - 30° - 45° = 105°
• Once all angles are known, the remaining side lengths can be calculated using the Law of Sines
  • Example: Using the triangle from the previous example, side a can be calculated as $a = \frac{c \sin(A)}{\sin(C)} = \frac{10 \sin(30°)}{\sin(105°)} \approx 5.7$

Solving oblique triangles with the Law of Sines

• The Law of Sines is particularly useful for solving oblique triangles, which are triangles that do not contain a right angle
• When given two sides and a non-included angle (SSA), the Law of Sines can be used to find the angle opposite one of the given sides, and then the third angle can be calculated using the angle sum property
  • Example: If side a is 8 units, side c is 12 units, and angle B is 40°, angle A can be calculated using the Law of Sines: $\sin(A) = \frac{a \sin(B)}{c} = \frac{8 \sin(40°)}{12} \approx 0.429$, so angle A is approximately 25.4°
  • The third angle, angle C, can then be found using the angle sum property: 180° - 40° - 25.4° ≈ 114.6°
• After all angles are known, the remaining side length can be calculated using the Law of Sines

The Ambiguous Case of the Law of Sines

Identifying the ambiguous case

• The ambiguous case in the Law of Sines occurs when given two sides and a non-included angle (SSA), and the given angle is not the largest angle in the triangle
• In the ambiguous case, there may be two possible solutions, one solution, or no solution for the triangle
• To determine the number of solutions, compare the length of the side opposite the given angle to the product of the sine of the given angle and the length of the other given side
  • If the opposite side is shorter than the product, there will be two solutions (one acute and one obtuse triangle)
  • If the opposite side is equal to the product, there will be one solution (a right triangle)
  • If the opposite side is longer than the product, there will be no solution

Solving triangles in the ambiguous case

• When two solutions exist, one triangle will have an obtuse angle opposite the longer given side, and the other will have an acute angle opposite the longer given side
• To find both solutions, calculate the angle opposite the shorter given side using the Law of Sines, and then find the third angle using the angle sum property
  • Example: If side a is 5 units, side c is 8 units, and angle B is 30°, the product of sin(B) and c is approximately 4 units, which is less than side a, indicating two solutions
  • For the acute solution, angle A can be found using the Law of Sines: $\sin(A) = \frac{a \sin(B)}{c} = \frac{5 \sin(30°)}{8} \approx 0.391$, so angle A is approximately 23°
  • For the obtuse solution, angle A can be found by subtracting the acute angle A from 180°: 180° - 23° = 157°
  • The third angle for each solution can be found using the angle sum property

Solving triangles with the Law of Cosines

Using the Law of Cosines with three sides

• The Law of Cosines is an extension of the Pythagorean Theorem for oblique triangles. It states that in any triangle with sides a, b, and c, and angles A, B, and C opposite their respective sides: $a^2 = b^2 + c^2 - 2bc \cos(A)$, $b^2 = a^2 + c^2 - 2ac \cos(B)$, and $c^2 = a^2 + b^2 - 2ab \cos(C)$
• When given all three side lengths of a triangle (SSS), the Law of Cosines can be used to calculate each of the triangle's angles
  • Example: If side a is 5 units, side b is 7 units, and side c is 9 units, angle A can be calculated using the Law of Cosines: $\cos(A) = \frac{b^2 + c^2 - a^2}{2bc} = \frac{7^2 + 9^2 - 5^2}{2 \cdot 7 \cdot 9} \approx 0.7063$, so angle A is approximately 45.2°
  • Angles B and C can be calculated similarly using the Law of Cosines
• The Law of Cosines is particularly useful when given three sides of a triangle, as the Law of Sines cannot be directly applied without at least one angle

Solving triangles with the Law of Cosines and two sides and an included angle

• When given two sides and their included angle (SAS), the Law of Cosines can be used to find the length of the third side
  • Example: If side a is 6 units, side b is 8 units, and angle C is 50°, side c can be calculated using the Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos(C) = 6^2 + 8^2 - 2 \cdot 6 \cdot 8 \cos(50°) \approx 81.9$, so side c is approximately 9.1 units
• After finding the third side, the remaining angles can be calculated using the Law of Sines or the angle sum property

Applications of the Law of Sines and Cosines

Solving real-world problems with oblique triangles

• Real-world problems involving oblique triangles can be solved by applying the Law of Sines, the Law of Cosines, or a combination of both, depending on the given information
• Common applications include navigation (determining distances and angles between locations), surveying (measuring land boundaries and elevations), engineering (designing structures and machines), and physics problems (analyzing forces and velocities)
• When solving real-world problems, it is essential to:
  1. Identify the given information
  2. Sketch a diagram of the triangle
  3. Determine which law is appropriate based on the given data
  4. Calculate the missing side lengths and angles
  5. Interpret the results within the context of the problem and check for reasonableness

Handling the ambiguous case in real-world problems

• In some cases, the ambiguous case may arise in real-world problems, and it is crucial to consider the practical implications of multiple solutions or no solution within the context of the problem
  • Example: When determining the location of a ship using two known distances and a bearing angle, the ambiguous case may indicate two possible locations, one solution (the ship is located along the line of sight), or no solution (the given information is inconsistent or impossible)
• In such situations, additional information or constraints may be needed to determine the most appropriate solution or to revise the problem's given information
• It is important to communicate the limitations and uncertainties associated with the ambiguous case when presenting solutions to real-world problems