Solution of Triangles: Cosine Rule

The cosine rule, also known as the law of cosines, is an important theorem in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. This rule is particularly useful in solving triangles, which means finding the unknown sides and angles of a triangle when certain other sides and angles are known.

The Cosine Rule Formula

The cosine rule states that for any triangle with sides of lengths a, b, and c, and angles opposite those sides A, B, and C respectively, the following relationship holds:

$$ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) $$

Similarly, the rule can be applied to the other angles of the triangle:

$$ a^2 = b^2 + c^2 - 2bc \cdot \cos(A) $$

$$ b^2 = a^2 + c^2 - 2ac \cdot \cos(B) $$

When to Use the Cosine Rule

The cosine rule is used in the following scenarios:

• To find the third side of a triangle when two sides and the included angle are known.
• To find the angles of a triangle when all three sides are known.

Differences and Important Points

Scenario	Sine Rule	Cosine Rule
Known Quantities	Two angles and one side, or two sides and a non-included angle	Two sides and the included angle, or all three sides
Type of Triangle	Any triangle	Any triangle
Formula	$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$	$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$
Solves for	Sides and angles	Sides and angles

Examples

Example 1: Finding the Third Side

Given a triangle with sides a = 5, b = 7, and angle C = 60°, find the length of side c.

Using the cosine rule:

$$ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) $$

$$ c^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot \cos(60°) $$

$$ c^2 = 25 + 49 - 70 \cdot \frac{1}{2} $$

$$ c^2 = 74 - 35 $$

$$ c^2 = 39 $$

$$ c = \sqrt{39} $$

$$ c \approx 6.24 $$

Example 2: Finding an Angle

Given a triangle with sides a = 4, b = 6, and c = 8, find angle C.

Using the cosine rule rearranged to solve for the angle:

$$ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} $$

$$ \cos(C) = \frac{4^2 + 6^2 - 8^2}{2 \cdot 4 \cdot 6} $$

$$ \cos(C) = \frac{16 + 36 - 64}{48} $$

$$ \cos(C) = \frac{-12}{48} $$

$$ \cos(C) = -\frac{1}{4} $$

To find angle C, we take the inverse cosine:

$$ C = \cos^{-1}\left(-\frac{1}{4}\right) $$

$$ C \approx 104.48° $$

Conclusion

The cosine rule is a versatile tool in solving triangles. It is essential to remember the conditions under which it is applicable and to be comfortable with rearranging the formula to solve for the desired quantity, whether it be a side length or an angle. With practice, the cosine rule becomes an invaluable part of a mathematician's toolkit for dealing with non-right-angled triangles.