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What Is the Law of Cosines?

Triangle with the law of cosines

The law of cosines is the general form of the Pythagorean theorem, which means it can be used on any triangle. You can see that when $A = 90^{\circ}$ , you get $- 2 b c \cdot \cos A = 0$ , and you’re left with the Pythagorean theorem.

Formula

The Law of Cosines

If $b$ and $c$ are two sides that span an angle $∠ A$ , then

\begin{align} a^{2} & = b^{2} + c^{2} - 2 b c \cdot \cos A & (1) \\ \cos A & = \frac{b^{2} + c^{2} - a^{2}}{2 b c} & (2) \end{align}

Rule

Uses

You can use the law of cosines to

Find a side in a triangle when you know the two other sides and one angle in the triangle—Formula (1).
Find an angle in a triangle when you know all three sides of the triangle—Formula (2).

Example 1

You have a quadrilateral $□ A B C D$ , where $A B = 12$ , $A D = 9$ and $∠ A = 120^{\circ}$ . Find the diagonal $B D$ .

The first thing to do is to make an auxiliary figure to help you visualize the problem.

Example of using the law of cosines 1

You can see that this fits with the law of cosines in (1). Insert the numbers to find $B D$ : $\begin{array}{l} B D^{2} & = A B^{2} + A D^{2} - 2 \cdot A B \cdot A D \cdot \cos 120^{\circ} \\ = 1 2^{2} + 9^{2} - 2 \cdot 12 \cdot 9 \cdot \cos 120^{\circ} \\ = 144 + 81 - 216 \cdot (- 0.5) \\ = 333 \\ B D & = \sqrt{333} \\ \approx 18.25 \end{array}$

Thus $B D \approx 18.25$ .

Example 2

You have a triangle $△ A B C$ , where $A B = 7$ , $A C = 5$ and $B C = 10$ . Find $∠ A$ .

The first thing to do is to make an auxiliary figure, to help you visualize the problem.

Example of using the law of cosines 2

You can see that this fits with the law of cosines in (2). Insert the numbers to find $∠ A$ . $\begin{array}{l} \cos A & = \frac{A B^{2} + A C^{2} - B C^{2}}{2 \cdot A B \cdot A C} \\ = \frac{7^{2} + 5^{2} - 1 0^{2}}{2 \cdot 7 \cdot 5} \\ = \frac{49 + 25 - 100}{70} \\ = \frac{- 26}{70} \\ A & = \cos^{- 1} (\frac{- 13}{35}) \\ \approx {111.8}^{\circ} \end{array}$

So the angle $∠ A$ is about $111.8$ $^{\circ}$ .

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