Law of Cosines

Law of Cosines

Before we begin, let us ease ourselves into this topic by recalling the Cosine Ratio and Pythagoras' Theorem. That way we can bridge the two concepts and derive the Law of Cosines as mentioned above.

Recap: The Cosine Ratio and Pythagoras' Theorem

Let us first recall the Pythagoras Theorem for a given right-angle triangle with an angle θ.

Right-angle triangle, Aishah Amri - StudySmarter Originals

Thus is given by,

\[c^2=a^2+b^2\].

Furthermore, recall that the cosine of an angle is found by dividing the adjacent by the hypotenuse. This is known as the Cosine Ratio

\[\cos(\theta=\frac{adjacent}{hypotenuse}=\frac{b}{c}\].

Derivation of the Law of Cosines

Now that we have set the ideas above, let us use them as a building block in establishing the Law of Cosines. Consider the triangle ABC below.

Law of Cosines, Aishah Amri - StudySmarter Originals

The triangle ABC is split into two triangles: ABD and BCD. Both these triangles are right triangles divided by the perpendicular leg BD of measure h. The side b is also divided into two sections AD = x and CD = b – x. Furthermore, the hypotenuse of triangle ABD is AB = c while the hypotenuse of triangle BCD is BC = a.

Let us analyse the relationship between the lengths of a, b, c and the angle A. By Pythagoras Theorem, we can solve triangle DBC by

Expanding (b - x)², we obtain

\[a^2=b^2-2bx+x^2+h^2\]

Similarly, we can do this for triangle ADB as

\[c^2=x^2+h^2\]

Substituting this expression for x² + h² into the previous equation, we obtain

\[a^2=b^2-2bx+c^2\]

Notice that we can further write x in the form of a Cosine Ratio as

\[\cos(A)=\frac{x}{c}\Rightarrow x=c\cdot \cos(A)\]

Replacing this expression for x, we obtain

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

Applying the Commutative Property and rearranging this, we obtain the Law of Cosines

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

as we have stated at the beginning of this section.

Variations of the Law of Cosines

The Law of Cosines has two other variations from the one given above. By the Law of Cosines, we see that the formula gives us the measure of one unknown side of a triangle. However, a triangle has three sides and three angles. In some cases, we may need to apply this concept on any of the other sides should we require to find their measures. Let us consider the triangle ABC below.

Variations of the Law of Cosines, Aishah Amri - StudySmarter Originals

Here, the value of a, b and c represent the length of each side of this triangle. Angles A, B and C describe the opposite angles for each of these sides respectively. There are three forms of the Law of Cosines for this triangle.

\begin{align} a^2&=b^2+c^2-2bc\cos(A)\\b^2&=a^2+c^2-2ac\cos(B)\\ c^2&=a^2+b^2-2ab\cos(A)\end{align}

The forms above are suitable for finding the length of an unknown side given the measures of two sides and their included angle. However, if we are given the values of three sides, we may need to perform some lengthy algebra to determine the unknown angle. To simplify such calculations, we can rearrange the expressions above so that we get an explicit formula for the unknown angle.

\begin{align} \cos(A)&=\frac{b^2+c^2-a^2}{2bc}\\\cos(B)&=\frac{a^2+c^2-b^2}{2ac} \\\cos(C)&=\frac{b^2+b^2-c^2}{2ab}\end{align}

Application of the Law of Cosines

In this section, we shall observe several worked examples that apply the Law of Cosines. We can apply the Law of Cosines for any triangle given the measures of two cases:

1. The value of two sides and their included angle

2. The value of three sides

Solving a Triangle Given Two Sides and their Included Angle

Solving a Triangle Given Three Sides

Real-World Example Involving Law of Cosines

Solving Oblique Triangles

An oblique triangle is one that has no right angle. Below are some examples of oblique triangles.

Oblique triangles, Aishah Amri - StudySmarter Originals

To solve such triangles, we need the length of at least one side and the value of any other two components of a given triangle. If the triangle has a solution, then we must choose whether to begin our solution by applying the Law of Sines or the Law of Cosines. The table below helps us decide on such situations.