Law of sines, Trigonometry of Triangles
Formula to find angle and side lengths
The law of sines provides a formula that relates the sides with the angles of a triangle. This formula allows you to relatively easily find the side length or the angle of any triangle.
When to use the law of sines formula.
You should use the law of sines when you know 2 sides and an angle (case 1 in the picture below) and you want to find the measure of an angle opposite a known side. Or when you know 2 angles and 1 side and want to get the side opposite a known angle (case 2 in picture below). In both cases, you must already know a side and an angle that are opposite of each other.
Cases when you can not use the Law of Sines
The picture below illustrates a case not suited for the law of sines. Since we do not know an opposite side and angle, we cannot employ the law of sines formula. (By the way, we could use the law of cosines to find the length of the side opposite the 115° angle. Once we know that side length ,we'd be able to use the law of sines as indicated by case 1 in the picture up above.)
Video on how to use the Law of SinesPractice Problems
Use the formula for law of sines to determine the measure of angle b to the nearest tenthDouble triangle Problems
Is it possible to use the law of sines to calculate x pictured in the triangle below?
So if you are given any 2 sides and an angle, can you always use the law of sines to find all the other sides and angles of a triangle?
The answer is Not always!
Consider the following. Let's imagine that we know that there is some 'triangle' ABC with the following information:
BC = 23 ,AC = 3, = 44. Below represents an accurate picture of these sides and angles. Red represents a given of the triangle (2 sides and 1 angle). Grey represents a variable that we can change (the grey 'side' AB). The gray line represents the third side of the triangle that we don't know. The only thing we know about that side is that it must extend at 44 degrees. As you can see from the picture, there is no way for that grey side to rise at 44 degrees and also to intersect with point A. Therefore no triangle can be drawn with these givens.
What is the length of side CB?