How To Solve A Non Right Triangle

Navigating the world of triangles often involves familiar territory: right triangles, with their neat 90-degree angles and straightforward trigonometric relationships. However, the landscape shifts when we venture into the realm of non-right triangles, also known as oblique triangles. These triangles lack the comforting presence of a right angle, demanding a different set of tools and techniques for solving them. This article provides a comprehensive guide on how to solve a non-right triangle, covering the necessary laws, formulas, and step-by-step methods.

Understanding Non-Right Triangles

Before diving into the solution methods, it's crucial to understand the characteristics of non-right triangles. A non-right triangle is any triangle that does not contain a right angle. This means all three angles are either acute (less than 90 degrees) or obtuse (one angle greater than 90 degrees and two acute angles). Solving a triangle means finding the measures of all three angles and the lengths of all three sides.

The key to solving non-right triangles lies in the Law of Sines and the Law of Cosines, which provide the necessary relationships between angles and sides. Let's explore these laws in detail.

The Law of Sines: A Powerful Tool

The Law of Sines establishes a relationship between the sides of a triangle and the sines of their opposite angles. In any triangle ABC, where a, b, and c are the lengths of the sides opposite angles A, B, and C respectively, the Law of Sines states:

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

This law is particularly useful when you know:

• Two angles and one side (AAS or ASA): You can find the remaining angle and the other two sides.
• Two sides and an angle opposite one of them (SSA): This is known as the ambiguous case, as it may lead to one, two, or no possible solutions.

Solving AAS and ASA Triangles Using the Law of Sines

Let's consider an example:

Given: Angle A = 30°, Angle B = 70°, Side a = 8 cm

Goal: Find Angle C, Side b, and Side c

Steps:

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

2. Find Side b: Using the Law of Sines, we can set up the following proportion:

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

   Substituting the given values:

   8 / sin(30°) = b / sin(70°)

   Solving for b:

   b = (8 * sin(70°)) / sin(30°) ≈ 15.04 cm

3. Find Side c: Again, using the Law of Sines:

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

   Substituting the values:

   8 / sin(30°) = c / sin(80°)

   Solving for c:

   c = (8 * sin(80°)) / sin(30°) ≈ 15.76 cm

Therefore, Angle C = 80°, Side b ≈ 15.04 cm, and Side c ≈ 15.76 cm.

The Ambiguous Case (SSA) and Its Challenges

The SSA case is called "ambiguous" because the given information might result in zero, one, or two possible triangles. This ambiguity arises from the fact that the given side opposite the angle might swing in two different positions, creating two valid triangles.

To determine the number of possible solutions, we need to compare the length of the side opposite the given angle (a) with the height (h) of the triangle, which can be calculated as h = b * sin(A), where b is the other given side and A is the given angle.

• If a < h: No triangle can be formed.
• If a = h: One right triangle can be formed.
• If h < a < b: Two possible triangles can be formed.
• If a ≥ b: One triangle can be formed.

Example of the Ambiguous Case:

Given: Angle A = 30°, Side a = 5 cm, Side b = 10 cm

Goal: Find Angle B, Angle C, and Side c

Steps:

1. Calculate the Height (h): h = b * sin(A) = 10 * sin(30°) = 5 cm

2. Analyze the Situation: Since a = h, one right triangle can be formed. This means Angle B = 90°.

3. Find Angle C: Angle C = 180° - Angle A - Angle B = 180° - 30° - 90° = 60°.

4. Find Side c: Using the Pythagorean theorem (since it's a right triangle):

   c = √(b² - a²) = √(10² - 5²) = √75 ≈ 8.66 cm

However, let's consider a different scenario:

Given: Angle A = 30°, Side a = 6 cm, Side b = 10 cm

Steps:

1. Calculate the Height (h): h = b * sin(A) = 10 * sin(30°) = 5 cm

2. Analyze the Situation: Since h < a < b, two possible triangles can be formed.

3. Find Angle B (First Possible Solution): Using the Law of Sines:

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

   6 / sin(30°) = 10 / sin(B)

   sin(B) = (10 * sin(30°)) / 6 ≈ 0.8333

   B = arcsin(0.8333) ≈ 56.44°

4. Find Angle C (First Possible Solution): Angle C = 180° - Angle A - Angle B = 180° - 30° - 56.44° ≈ 93.56°

5. Find Side c (First Possible Solution): Using the Law of Sines:

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

   6 / sin(30°) = c / sin(93.56°)

   c = (6 * sin(93.56°)) / sin(30°) ≈ 11.98 cm

6. Find Angle B (Second Possible Solution): The second possible angle B is the supplement of the first solution:

   B' = 180° - B = 180° - 56.44° ≈ 123.56°

7. Find Angle C (Second Possible Solution): Angle C' = 180° - Angle A - Angle B' = 180° - 30° - 123.56° ≈ 26.44°

8. Find Side c (Second Possible Solution): Using the Law of Sines:

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

   6 / sin(30°) = c / sin(26.44°)

   c = (6 * sin(26.44°)) / sin(30°) ≈ 5.32 cm

In this case, we have two possible triangles:

• Triangle 1: Angle B ≈ 56.44°, Angle C ≈ 93.56°, Side c ≈ 11.98 cm
• Triangle 2: Angle B ≈ 123.56°, Angle C ≈ 26.44°, Side c ≈ 5.32 cm

This example highlights the importance of carefully analyzing the SSA case and considering all possible solutions.

The Law of Cosines: Tackling SAS and SSS Triangles

The Law of Cosines provides a relationship between the sides and angles of a triangle that is particularly useful when you know:

• Two sides and the included angle (SAS): You can find the remaining side.
• Three sides (SSS): You can find all three angles.

The Law of Cosines consists of three formulas:

• a² = b² + c² - 2bc * cos(A)
• b² = a² + c² - 2ac * cos(B)
• c² = a² + b² - 2ab * cos(C)

Solving SAS Triangles Using the Law of Cosines

Let's consider an example:

Given: Side a = 10 cm, Side b = 12 cm, Angle C = 40°

Goal: Find Side c, Angle A, and Angle B

Steps:

1. Find Side c: Using the Law of Cosines:

   c² = a² + b² - 2ab * cos(C)

   Substituting the given values:

   c² = 10² + 12² - 2 * 10 * 12 * cos(40°)

   c² = 100 + 144 - 240 * cos(40°) ≈ 59.03

   c = √59.03 ≈ 7.68 cm

2. Find Angle A: Now, we can use either the Law of Sines or the Law of Cosines. Let's use the Law of Sines:

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

   10 / sin(A) = 7.68 / sin(40°)

   sin(A) = (10 * sin(40°)) / 7.68 ≈ 0.8367

   A = arcsin(0.8367) ≈ 56.81°

3. Find Angle B: Angle B = 180° - Angle A - Angle C = 180° - 56.81° - 40° ≈ 83.19°

Therefore, Side c ≈ 7.68 cm, Angle A ≈ 56.81°, and Angle B ≈ 83.19°.

Solving SSS Triangles Using the Law of Cosines

Let's consider an example:

Given: Side a = 8 cm, Side b = 5 cm, Side c = 7 cm

Goal: Find Angle A, Angle B, and Angle C

Steps:

1. Find Angle A: Using the Law of Cosines:

   a² = b² + c² - 2bc * cos(A)

   Rearranging to solve for cos(A):

   cos(A) = (b² + c² - a²) / (2bc)

   Substituting the given values:

   cos(A) = (5² + 7² - 8²) / (2 * 5 * 7) = (25 + 49 - 64) / 70 = 10 / 70 = 1/7

   A = arccos(1/7) ≈ 81.79°

2. Find Angle B: Using the Law of Cosines:

   b² = a² + c² - 2ac * cos(B)

   Rearranging to solve for cos(B):

   cos(B) = (a² + c² - b²) / (2ac)

   Substituting the given values:

   cos(B) = (8² + 7² - 5²) / (2 * 8 * 7) = (64 + 49 - 25) / 112 = 88 / 112 = 11/14

   B = arccos(11/14) ≈ 38.21°

3. Find Angle C: Angle C = 180° - Angle A - Angle B = 180° - 81.79° - 38.21° = 60°

Therefore, Angle A ≈ 81.79°, Angle B ≈ 38.21°, and Angle C = 60°.

Heron's Formula: Finding the Area of a Triangle

While not directly used for solving a triangle (finding angles and sides), Heron's formula is a useful tool for finding the area of a triangle when you know the lengths of all three sides (SSS).

Heron's Formula:

Area = √(s(s - a)(s - b)(s - c))

Where a, b, and c are the lengths of the sides, and s is the semi-perimeter, calculated as:

s = (a + b + c) / 2

Example:

Using the previous SSS example where a = 8 cm, b = 5 cm, and c = 7 cm:

1. Calculate the Semi-Perimeter (s): s = (8 + 5 + 7) / 2 = 10 cm

2. Apply Heron's Formula:

   Area = √(10(10 - 8)(10 - 5)(10 - 7)) = √(10 * 2 * 5 * 3) = √300 ≈ 17.32 cm²

Therefore, the area of the triangle is approximately 17.32 cm².

Choosing the Right Law: A Strategic Approach

Selecting the appropriate law for solving a non-right triangle is crucial for efficiency and accuracy. Here's a quick guide:

• Law of Sines: Use when you know two angles and a side (AAS or ASA), or two sides and an angle opposite one of them (SSA - be mindful of the ambiguous case!).
• Law of Cosines: Use when you know two sides and the included angle (SAS) or three sides (SSS).

In some cases, you might need to use both laws to fully solve a triangle. For instance, you might use the Law of Cosines to find a side and then use the Law of Sines to find an angle.

Practical Applications

The ability to solve non-right triangles has numerous practical applications in various fields, including:

• Surveying: Determining distances and angles in land measurement.
• Navigation: Calculating distances and bearings in air and sea travel.
• Engineering: Designing structures and calculating forces.
• Physics: Analyzing projectile motion and vector components.
• Astronomy: Determining distances to stars and planets.

Conclusion

Solving non-right triangles requires a solid understanding of the Law of Sines and the Law of Cosines. By mastering these laws and carefully analyzing the given information, you can confidently tackle a wide range of triangle-related problems. Remember to be particularly cautious when dealing with the ambiguous case (SSA) and always consider all possible solutions. With practice and a strategic approach, you'll be well-equipped to navigate the world of oblique triangles.