The law of cosines can be used on a triangle to find the length of an unknown side under certain conditions. Let’s explore these conditions step-by-step:
1. Two Sides and the Included Angle are Known:
- If you know the lengths of two sides of a triangle and the measure of the angle between those two sides (the included angle), you can use the law of cosines to find the length of the third side.
- The formula for the law of cosines is:
[tex]\[
a^2 = b^2 + c^2 - 2bc \cos(A)
\][/tex]
where [tex]\( a \)[/tex] is the side opposite the included angle [tex]\( A \)[/tex], and [tex]\( b \)[/tex] and [tex]\( c \)[/tex] are the other two sides.
For example, if you know [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( A \)[/tex], you can find [tex]\( a \)[/tex] using the above formula.
2. All Three Sides are Known:
- If you know the lengths of all three sides of a triangle, you can use the law of cosines to find any of the angles.
- Rewriting the law of cosines to solve for an angle, the formula is:
[tex]\[
\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}
\][/tex]
You can solve for [tex]\( A \)[/tex] by taking the inverse cosine (arccos) of the right-hand side.
In summary, the law of cosines can be applied to any triangle where:
- You know the lengths of two sides and the measure of the included angle (to find the third side), or
- You know the lengths of all three sides (to find any of the angles).
These conditions are not limited to any specific type of triangle; they can be applied so long as the aforementioned information is available.
