The Law of Cosines can be applied to any triangle, regardless of its type—whether it is acute, obtuse, or right. It is particularly useful in situations where:
1. We know the lengths of two sides of the triangle, and we know the measure of the included angle (the angle between the two known sides). By using the Law of Cosines, we can then find the length of the third, unknown side.
The general form of the Law of Cosines for a side [tex]\(a\)[/tex] opposite an angle [tex]\(A\)[/tex] is given by:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
Here is a step-by-step outline of how the Law of Cosines can be used:
### Step-by-Step Solution:
1. Identify the Known Values:
- Let [tex]\( b \)[/tex] and [tex]\( c \)[/tex] be the lengths of the two known sides.
- Let [tex]\( A \)[/tex] be the measure of the known angle between these two sides.
- Let [tex]\( a \)[/tex] be the length of the side opposite the angle [tex]\( A \)[/tex] which we need to find.
2. Write Down the Law of Cosines:
- The formula to use is:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
3. Plug in the Known Values:
- Substitute the lengths [tex]\( b \)[/tex] and [tex]\( c \)[/tex] and the cosine of angle [tex]\( A \)[/tex] into the Law of Cosines formula.
4. Solve for [tex]\( a^2 \)[/tex]:
- Calculate the value on the right-hand side of the equation to find [tex]\( a^2 \)[/tex].
5. Find the Length of [tex]\( a \)[/tex]:
- Take the square root of both sides of the equation to solve for [tex]\( a \)[/tex].
[tex]\[ a = \sqrt{b^2 + c^2 - 2bc \cos(A)} \][/tex]
### Conclusion:
The Law of Cosines can be used to find the length of an unknown side in any triangle where we know two sides and the included angle. This makes it a powerful tool in solving triangle-related problems in a wide array of contexts, not just limited to right triangles.
By following the steps outlined, you can determine the length of the unknown side given the appropriate known values.
