When To Use Law Of Sines Vs Law Of Cosines

In trigonometry, the Law of Sines and the Law of Cosines are powerful tools for solving triangles, especially when dealing with non-right triangles. Knowing when to apply each law is crucial for accurate calculations and problem-solving. This article provides a comprehensive guide on understanding and differentiating between these two laws, ensuring you choose the right approach every time.

Understanding the Law of Sines

The Law of Sines establishes a relationship between the angles of a triangle and the lengths of their opposite sides. It's particularly useful when you have information about angles and sides that are opposite each other.

The Formula

The Law of Sines is expressed as follows:

a / sin(A) = b / sin(B) = c / sin(C)

Where:

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

When to Use the Law of Sines

The Law of Sines is best applied in the following scenarios:

1. Angle-Side-Angle (ASA): You know two angles and the included side (the side between the two angles).

2. Angle-Angle-Side (AAS): You know two angles and a non-included side (a side not between the two angles).

3. Side-Side-Angle (SSA): You know two sides and an angle opposite one of those sides. This case is known as the ambiguous case and requires extra caution because it can lead to zero, one, or two possible triangle solutions.

Example: Solving a Triangle with ASA

Suppose we have a triangle where angle A = 30°, angle B = 70°, and side c = 10 units. We want to find the remaining angles and sides.

1. Find angle C: Since the sum of angles in a triangle is 180°, C = 180° - A - B = 180° - 30° - 70° = 80°.

2. Apply the Law of Sines to find side a:

   a / sin(A) = c / sin(C)

   a / sin(30°) = 10 / sin(80°)

   a = (10 * sin(30°)) / sin(80°) ≈ 5.077 units.

3. Apply the Law of Sines to find side b:

   b / sin(B) = c / sin(C)

   b / sin(70°) = 10 / sin(80°)

   b = (10 * sin(70°)) / sin(80°) ≈ 9.540 units.

Understanding the Law of Cosines

The Law of Cosines is another fundamental trigonometric law that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is particularly useful when you do not have a direct angle-side relationship, such as when you know all three sides or two sides and the included angle.

The Formula

The Law of Cosines can be expressed in three forms, each solving for a different side or angle:

1. a² = b² + c² - 2bc * cos(A)

2. b² = a² + c² - 2ac * cos(B)

3. c² = a² + b² - 2ab * cos(C)

Where:

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

When to Use the Law of Cosines

The Law of Cosines is best applied in the following scenarios:

1. Side-Angle-Side (SAS): You know two sides and the included angle (the angle between the two sides).

2. Side-Side-Side (SSS): You know all three sides of the triangle.

Example: Solving a Triangle with SAS

Suppose we have a triangle where side a = 5 units, side b = 8 units, and angle C = 77°. We want to find the remaining angles and sides.

1. Apply the Law of Cosines to find side c:

   c² = a² + b² - 2ab * cos(C)

   c² = 5² + 8² - 2 * 5 * 8 * cos(77°)

   c² = 25 + 64 - 80 * cos(77°)

   c² ≈ 25 + 64 - 80 * 0.225

   c² ≈ 89 - 18

   c² ≈ 71

   c ≈ √71 ≈ 8.426 units.

2. Apply the Law of Cosines to find angle A:

   a² = b² + c² - 2bc * cos(A)

   5² = 8² + (√71)² - 2 * 8 * √71 * cos(A)

   25 = 64 + 71 - 16√71 * cos(A)

   25 = 135 - 16√71 * cos(A)

   -110 = -16√71 * cos(A)

   cos(A) = 110 / (16√71)

   A = arccos(110 / (16√71)) ≈ 27.66°.

3. Find angle B:

   B = 180° - A - C

   B = 180° - 27.66° - 77°

   B ≈ 75.34°.

Example: Solving a Triangle with SSS

Suppose we have a triangle where side a = 7 units, side b = 9 units, and side c = 5 units. We want to find the angles A, B, and C.

1. Apply the Law of Cosines to find angle A:

   a² = b² + c² - 2bc * cos(A)

   7² = 9² + 5² - 2 * 9 * 5 * cos(A)

   49 = 81 + 25 - 90 * cos(A)

   49 = 106 - 90 * cos(A)

   -57 = -90 * cos(A)

   cos(A) = 57 / 90

   A = arccos(57 / 90) ≈ 50.70°.

2. Apply the Law of Cosines to find angle B:

   b² = a² + c² - 2ac * cos(B)

   9² = 7² + 5² - 2 * 7 * 5 * cos(B)

   81 = 49 + 25 - 70 * cos(B)

   81 = 74 - 70 * cos(B)

   7 = -70 * cos(B)

   cos(B) = -7 / 70 = -0.1

   B = arccos(-0.1) ≈ 95.74°.

3. Find angle C:

   C = 180° - A - B

   C = 180° - 50.70° - 95.74°

   C ≈ 33.56°.

Key Differences and How to Choose

To summarize, here's a quick guide on when to use each law:

• Law of Sines: Use when you have an angle and its opposite side (or when you can easily find one). Specifically, use it for ASA, AAS, or SSA cases. Remember to be cautious with the ambiguous SSA case.
• Law of Cosines: Use when you do not have an angle and its opposite side. Specifically, use it for SAS or SSS cases.

Here's a table summarizing the conditions:

The Ambiguous Case (SSA) in Detail

The ambiguous case (SSA) arises when you are given two sides and a non-included angle. This case can result in zero, one, or two possible triangles. To determine the number of possible solutions, you need to analyze the given information carefully.

Let's say you are given side a, side b, and angle A.

1. If a < h (h is the height of the triangle from vertex C to side c): No triangle exists.

2. If a = h: One right triangle exists.

3. If a > h and a < b: Two possible triangles exist.

4. If a ≥ b: One triangle exists.

Where h can be calculated as h = b * sin(A).

Example of the Ambiguous Case

Suppose a triangle has side a = 20, side b = 30, and angle A = 30°. Determine the number of possible triangles.

1. Calculate h:

   h = b * sin(A) = 30 * sin(30°) = 30 * 0.5 = 15.

2. Compare a and h:

   Since a = 20 and h = 15, a > h.

3. Compare a and b:

   Since a = 20 and b = 30, a < b.

4. Conclusion:

   Because a > h and a < b, there are two possible triangles.

When solving the ambiguous case, be prepared to find two sets of possible solutions, each corresponding to a valid triangle.

Practical Tips and Considerations

1. Draw a Diagram: Always start by drawing a diagram of the triangle. Label the known sides and angles. This visual representation can help you identify the correct law to use and avoid errors.

2. Check for Obtuse Angles: If you are using the Law of Cosines to find an angle and the cosine value is negative, remember that the angle is obtuse (greater than 90°).

3. Use the Law of Sines After Law of Cosines: If you use the Law of Cosines to find a side or angle, and then need to find another angle, it's generally safer to use the Law of Cosines again to avoid the ambiguous case of the Law of Sines.

4. Accuracy: Maintain as much precision as possible throughout your calculations. Rounding intermediate values can lead to significant errors in the final answer.

5. Units: Ensure that all measurements are in the same units (e.g., all lengths in meters, all angles in degrees).

Advanced Applications and Real-World Examples

The Law of Sines and the Law of Cosines are not just theoretical concepts; they have numerous practical applications in various fields.

1. Navigation: Sailors and pilots use these laws to calculate distances and bearings. By knowing the angles between landmarks and the distances to some landmarks, they can determine their position.

2. Surveying: Surveyors use these laws to measure land areas and create accurate maps. They can determine distances and angles that are difficult or impossible to measure directly.

3. Engineering: Engineers use these laws in structural analysis to calculate forces and stresses in bridges, buildings, and other structures.

4. Astronomy: Astronomers use these laws to determine distances to stars and planets. By measuring the angles between celestial objects and applying trigonometric principles, they can estimate distances in space.

5. Physics: Physicists use these laws in mechanics to analyze the motion of objects. They can calculate forces and velocities in various scenarios involving triangles and angles.

Example: Navigation

A ship sails from point A to point B, a distance of 50 nautical miles. It then turns 60° towards point C, a distance of 80 nautical miles. Find the distance from point A to point C.

1. Identify the Knowns:

   • Side AB = 50 nautical miles
   • Side BC = 80 nautical miles
   • Angle B = 180° - 60° = 120° (since the ship turned 60°, the interior angle at B is the supplement of 60°)

2. Apply the Law of Cosines:

   AC² = AB² + BC² - 2 * AB * BC * cos(B)

   AC² = 50² + 80² - 2 * 50 * 80 * cos(120°)

   AC² = 2500 + 6400 - 8000 * (-0.5)

   AC² = 8900 + 4000

   AC² = 12900

   AC = √12900 ≈ 113.58 nautical miles

Therefore, the distance from point A to point C is approximately 113.58 nautical miles.

Common Mistakes to Avoid

1. Misidentifying the Case: The most common mistake is using the wrong law. Always carefully identify whether you have ASA, AAS, SSA, SAS, or SSS.

2. Ambiguous Case Oversights: Failing to recognize or correctly handle the ambiguous case (SSA) can lead to incorrect solutions.

3. Incorrect Angle Measures: Ensure that angles are measured in the correct units (degrees or radians) and that you are using the correct angle in the formula.

4. Rounding Errors: Rounding intermediate calculations can introduce errors. Keep as many decimal places as possible until the final answer.

5. Algebraic Errors: Be careful with algebraic manipulations, especially when rearranging equations to solve for unknown variables.

Conclusion

Mastering the Law of Sines and the Law of Cosines is essential for anyone studying trigonometry or working in fields that require trigonometric calculations. Understanding when to apply each law, recognizing the ambiguous case, and avoiding common mistakes will enable you to solve a wide range of problems accurately and efficiently. By practicing with various examples and real-world applications, you can develop a strong foundation in trigonometry and enhance your problem-solving skills. Remember to always start with a clear diagram, carefully identify the given information, and choose the appropriate law based on the specific case. With diligent practice and a solid understanding of these principles, you will be well-equipped to tackle any trigonometric challenge.