COSINE LAW | LAW OF COSINES | COSINE FORMULA | Math 9 | Trigonometry | OBLIQUE TRIANGLES

Key Concepts

• Oblique Triangles
• Law of Cosines (Cosine Law/Cosine Formula/Cosine Rule)
• SAS (Side-Angle-Side) Pattern
• SSS (Side-Side-Side) Pattern
• Arc Cosine (Inverse Cosine)

Cosine Law: Introduction and Formulas

The video explains the Law of Cosines, a method used to solve oblique triangles (triangles without a right angle). Oblique triangles can be either acute or obtuse. Unlike right triangles, which can be solved using trigonometric ratios (SOH CAH TOA) and the Pythagorean theorem, oblique triangles require the Law of Sines or the Law of Cosines.

The Law of Cosines is stated as: "the square of the length of one side is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the angle between them."

Given a triangle ABC, where side 'a' is opposite angle A, side 'b' is opposite angle B, and side 'c' is opposite angle C, the formulas for finding the sides are:

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

These formulas are used when given two sides and the included angle (SAS).

Cosine Law: Finding Angles

The video also provides formulas for finding the angles when all three sides are known (SSS). These are derived from the side formulas:

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

To find the angle itself, you need to take the arc cosine (inverse cosine) of the result.

When to Use Cosine Law

The Cosine Law is used in two specific scenarios:

1. SAS (Side-Angle-Side): When two sides and the included angle are known.
2. SSS (Side-Side-Side): When all three sides are known.

Example 1: Finding a Side (SAS)

The video demonstrates how to find the length of a side using the Cosine Law when given two sides and the included angle (SAS).

Problem: Given a triangle ABC with side c = 8, side b = 5, and angle A = 130 degrees, find the length of side a (represented as 'x').

Solution:

1. Identify the formula: Since we're looking for side 'a', we use the formula: a² = b² + c² - 2bc * cos(A)
2. Substitute the values: a² = 5² + 8² - 2 * 5 * 8 * cos(130°)
3. Solve for 'a': a = √(5² + 8² - 2 * 5 * 8 * cos(130°))
4. Calculate: a ≈ 11.85 (rounded to the nearest hundredth)

Therefore, the length of side 'a' (or 'x') is approximately 11.85.

Finding Remaining Angles:

The video then shows how to find the remaining angles (B and C) after finding side 'a'.

1. Find angle C: Use the formula cos(C) = (a² + b² - c²) / (2ab)
   • Substitute: cos(C) = (11.85² + 5² - 8²) / (2 * 11.85 * 5) ≈ 0.8559
   • Find arc cosine: C = arccos(0.8559) ≈ 31.14 degrees
2. Find angle B: Since the sum of angles in a triangle is 180 degrees, B = 180 - (A + C)
   • Substitute: B = 180 - (130 + 31.14) ≈ 18.86 degrees

Example 2: Finding an Angle (SSS)

The video demonstrates how to find an angle using the Cosine Law when given all three sides (SSS).

Problem: Given a triangle ABC with side a = 6, side b = 5, and side c = 10, find the measure of angle C (represented as 'x').

Solution:

1. Identify the formula: Since we're looking for angle 'C', we use the formula: cos(C) = (a² + b² - c²) / (2ab)
2. Substitute the values: cos(C) = (6² + 5² - 10²) / (2 * 6 * 5)
3. Calculate: cos(C) = (36 + 25 - 100) / 60 = -39 / 60 = -0.65
4. Find arc cosine: C = arccos(-0.65) ≈ 130.54 degrees

Therefore, the measure of angle 'C' (or 'x') is approximately 130.54 degrees. Since this angle is greater than 90 degrees, the triangle is an obtuse triangle.

Finding Remaining Angles:

1. Find angle A: Use the formula cos(A) = (b² + c² - a²) / (2bc)
   • Substitute: cos(A) = (5² + 10² - 6²) / (2 * 5 * 10) = 89 / 100 = 0.89
   • Find arc cosine: A = arccos(0.89) ≈ 27.13 degrees
2. Find angle B: Since the sum of angles in a triangle is 180 degrees, B = 180 - (A + C)
   • Substitute: B = 180 - (27.13 + 130.54) ≈ 22.33 degrees

Conclusion

The Law of Cosines is a valuable tool for solving oblique triangles when given either two sides and the included angle (SAS) or all three sides (SSS). The video provides clear formulas and step-by-step examples for both scenarios, including how to find the remaining angles after applying the Cosine Law. The video emphasizes the importance of using arc cosine to find the angle and rounding the answer to the nearest hundredth as instructed. The presenter concludes by mentioning that the next topic will be word problems involving oblique triangles.