The Law of Cosines is used to find unknown sides or angles in a triangle, particularly in non-right triangles. Here’s a sample problem, followed by its solution.

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Problem

A triangle has sides $a = 7$, $b = 10$, and an included angle $C = 60^\circ$. Find the length of side $c$.

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Solution

The Law of Cosines states: $c^2 = a^2 + b^2 - 2ab \cos(C)$

Step 1: Substitute the known values into the formula. $c^2 = 7^2 + 10^2 - 2(7)(10)\cos(60^\circ)$ $c^2 = 49 + 100 - 2(7)(10)(0.5)$ $c^2 = 49 + 100 - 70$ $c^2 = 79$

Step 2: Take the square root of both sides to find $c$. $c = \sqrt{79}$

Step 3: Approximate the result (if necessary). $c \approx 8.89$

Thus, the length of side $c$ is approximately 8.89 units.

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Do you want a detailed breakdown of any part of the calculation? Let me know!

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Follow-up Questions

1. How is the Law of Cosines derived?
2. Can the Law of Cosines be used to find angles instead of sides?
3. What happens to the Law of Cosines in a right triangle?
4. How does the Law of Cosines relate to the Pythagorean theorem?
5. Can you apply the Law of Cosines to solve a triangle with sides $8, 15,$ and $17$?

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Tip

Always verify whether you need to solve for a side or an angle. For angles, you’ll use the rearranged form of the Law of Cosines:
$\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$