The law of cosines can't be used when which of the following combinations of information is given?
The law of cosines is a formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It can be used to find the length of a side or the measure of an angle in a triangle when you have enough information. However, there are situations where the law of cosines cannot be directly applied. These situations occur when certain combinations of information are given.
Here are the combinations of information that would make the law of cosines not applicable:
1. When you have only the lengths of two sides: The law of cosines requires knowing the lengths of at least three sides of a triangle to be applied because it involves comparing the lengths of these sides.
2. When you have the lengths of all three sides but not enough information about the angles: To apply the law of cosines, you need to know either one angle and the lengths of the two adjacent sides, or two angles and the length of the side between them. If information is missing about the angles, the law of cosines cannot be used directly.
3. When you have the lengths of all three sides and information about their corresponding angles, but the triangle does not meet the triangle inequality theorem: The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If the triangle inequality theorem is violated, the law of cosines is not applicable.
In summary, the law of cosines cannot be used when you have only two side lengths, when you do not have sufficient information about the angles, or when the triangle does not satisfy the triangle inequality theorem.