For the most part, will a law of cosines always be one triangle? As in one triangle to solve?
yes. The reason the law of sines can give two triangles is because sin(x) is positive all the way from 0 to 180.
cos(x) becomes negative for x>90, so the formula takes that into account, always leaving only one possible answer.
I mean, think about it geometrically. If you know the lengths of two sides, and the angle between them, there's only one line segment that connects the two endpoints.
The Law of Cosines can be used to solve a triangle when you have enough information about its sides and angles. It applies to any triangle, whether it is acute (all angles < 90 degrees), obtuse (one angle > 90 degrees), or right-angled (one angle = 90 degrees). So, in theory, the Law of Cosines can be applied to any triangle you encounter.
However, it is important to note that the Law of Cosines involves calculating the cosine of an angle, which requires knowledge of the length of at least two sides of a triangle and the size of the angle between them. This means that the Law of Cosines alone may not always be suitable to solve a triangle with incomplete or insufficient information.
Additionally, the Law of Cosines is most commonly used when dealing with triangles that are not right-angled, as the Pythagorean Theorem can be used in such cases. In right-angled triangles, where one angle measures 90 degrees, the Pythagorean Theorem is often simpler and more appropriate for finding missing side lengths.
In summary, while the Law of Cosines can be applied to any triangle, it may not always be the most efficient or necessary method, particularly when dealing with right-angled triangles. Other trigonometric identities, such as the Pythagorean Theorem, may be more useful depending on the context of the problem.