-2bc·cos(A), to account for the fact that the triangle is not a right triangle.
We can write three versions of the LOC, one for every angle/opposite side pair: Use the law of cosines to find the missing measurements of the triangles in these two examples.
We are given the hypotenuse and need to find the adjacent side.
This formula which connects these three is: cos(angle) = adjacent / hypotenuse therefore, cos60 = x / 13 therefore, x = 13 × cos60 = 6.5 therefore the length of side x is 6.5cm. The following graphs show the value of sinø, cosø and tanø against ø (ø represents an angle).
In the first, the measures of two sides and the included angle (the angle between them) are known. : Set up the law of cosines using the only set of angles and sides for which it is possible in this case: Now using the new side, find one of the missing angles using the law of sines: And then the third angle is In general, try to use the law of sines first. But in this case, that wasn't possible, so the law of cosines was necessary.
: Set up the law of cosines to solve for either one of the angles: Rearrange to solve for A. This labeling scheme is commonly used for non-right triangles. Notice that we need to know at least one angle-opposite side pair for the Law of Sines to work.Capital letters are angles and the corresponding lower-case letters go with the side opposite the angle: side a (with length of a units) is across from angle A (with a measure of A degrees or radians), and so on. if we find the sines of angle A and angle C using their corresponding right triangles, we notice that they both contain the altitude, x. The law of sines can be used to find the measure of an angle or a side of a non-right triangle if we know: Use the law of sines to find the missing measurements of the triangles in these examples. : The missing angle is easy, it's just Now set up one of the law of sines proportions and solve for the missing piece, in this case the length of the lower side: Then do the same for the other missing side.After the law of cosines is applied to a triangle, the resulting information will always make it possible to use the law of sines to calculate further properties of the triangle.Consider another non-right triangle, labeled as shown with side lengths x and y.Check for ambiguities unless you're clear about the results from other clues in the problem.The law of cosines is computationally a little more complicated to use than the law of sines, but fortunately, it only needs to be used once.The sine equations are We can rearrange those by solving each for x (multiply by c on both sides of the left equation, and by a on both sides of the right): Now the transitive property says that if both c·sin(A) and a·sin(C) are equal to x, then they must be equal to each other: We usually divide both sides by ac to get the easy-to-remember expression of the law of sines: We could do the same derivation with the other two altitudes, drawn from angles A and C to come up with similar relations for the other angle pairs. It's best to use the original known angle and side so that round-off errors or mistakes don't add up.: This time we have to find our first missing angle using the LOS, then we can use the sum of the angles of a triangle to find the third. We'll be solving for a missing angle, so we'll have to calculate an inverse sine: Now it's easy to calculate the third angle: Then apply the law of sines again for the missing side.The two possible angles are always related: their sum is 180˚. Here are the two possible triangles in this example: In such a case, we have to know a little more about our triangle. In some cases, the quantity (180˚ minus our calculated angle) added back to the calculated angle will be greater than 180˚, and such a triangle cannot exist, so the solution is unique.Make sure to be aware of this as you work problems.