Here are three examples using the Law of Cosines for solving different types of triangle problems:

Example 1: Finding the Side of a Triangle (Given SAS - Side-Angle-Side)

Suppose you have a triangle with sides $b = 7$ units and $c = 5$ units, and the angle $A$ between them is $60^\circ$. Find the length of side $a$.

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

Substitute the values: $a^2 = 7^2 + 5^2 - 2 \cdot 7 \cdot 5 \cdot \cos(60^\circ)$ $a^2 = 49 + 25 - 70 \cdot 0.5$ $a^2 = 74 - 35 = 39$ $a = \sqrt{39} \approx 6.24 \, \text{units}$

Example 2: Finding the Angle of a Triangle (Given SSS - Side-Side-Side)

In a triangle with sides $a = 8$, $b = 6$, and $c = 10$, find the angle $A$.

Using the formula: $\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}$

Substitute the values: $\cos(A) = \frac{6^2 + 10^2 - 8^2}{2 \cdot 6 \cdot 10}$ $\cos(A) = \frac{36 + 100 - 64}{120}$ $\cos(A) = \frac{72}{120} = 0.6$ $A = \cos^{-1}(0.6) \approx 53.13^\circ$

Example 3: Finding the Side of a Triangle (Given SAS - Side-Angle-Side)

Consider a triangle with side $a = 12$ units, side $b = 15$ units, and an included angle $C = 45^\circ$. Find the length of side $c$.

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

Substitute the values: $c^2 = 12^2 + 15^2 - 2 \cdot 12 \cdot 15 \cdot \cos(45^\circ)$ $c^2 = 144 + 225 - 360 \cdot 0.7071$ $c^2 = 369 - 254.56 \approx 114.44$ $c = \sqrt{114.44} \approx 10.7 \, \text{units}$

Would you like further explanations on any of these examples or have other questions?

Related Questions:

1. How is the Law of Cosines used to determine angles in obtuse triangles?
2. What are some real-life applications of the Law of Cosines?
3. How can the Law of Cosines help in solving non-right triangles?
4. Can you use the Law of Cosines to prove the Pythagorean Theorem?
5. What’s the difference between using the Law of Cosines and the Law of Sines?

Tip:

The Law of Cosines is particularly useful when you know two sides and the included angle (SAS) or all three sides (SSS) of a triangle, making it versatile for solving non-right triangles.