How to Rearrange the Cosine Rule?

The cosine rule can be rearranged to solve for different unknowns in a triangle. Here's how to rearrange the cosine rule for different purposes.

The Cosine Rule

The standard cosine rule is:

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

Where:

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

Rearranging to Find an Angle

Often, you need to find an angle when you know all three sides. To do this, rearrange the cosine rule. Let's rearrange to find angle A:

1. Start with the standard formula: a² = b² + c² - 2bc cos(A)

2. Isolate the term with the cosine: 2bc cos(A) = b² + c² - a² (Add 2bc cos(A) to both sides and subtract a² from both sides)

3. Solve for cos(A): cos(A) = (b² + c² - a²) / (2bc)

4. Find the angle A: A = arccos((b² + c² - a²) / (2bc)) (Use the inverse cosine function, also written as cos^-1)

Similarly, for angles B and C:

• B = arccos((a² + c² - b²) / (2ac))
• C = arccos((a² + b² - c²) / (2ab))

Example:

If a = 5, b = 7, and c = 8, then to find angle A:

A = arccos((7² + 8² - 5²) / (2 7 8))
A = arccos((49 + 64 - 25) / 112)
A = arccos(88 / 112)
A ≈ 38.6 degrees

Rearranging to Find a Side (when you know an angle opposite to that side)

If you want to find a side, say 'a', and you know angle 'A', the formula is already in a convenient format:

a² = b² + c² - 2bc cos(A)

You can then simply take the square root to find 'a':

a = √(b² + c² - 2bc cos(A))

Example:

If b = 7, c = 8, and A = 38.6 degrees, then:

a = √(7² + 8² - 2 7 8 cos(38.6))
a = √(49 + 64 - 112 0.78)
a = √(113 - 87.36)
a ≈ 5.06

Summary

The cosine rule is a versatile tool for solving triangles. By rearranging the formula, you can find unknown sides or angles, given sufficient information. Remember to use the inverse cosine function (arccos or cos^-1) when solving for angles.