Solving triangles is an exercise in which we, with information about three of six attributes of a triangle (three sides and three angles), determine the other three attributes, thus knowing everything about the triangle. Note: This is not possible if the triangle is acute (all angles less than 90°) and if there are two sides given and an opposite angle, called the SSA case.
The Law of Sines
The Law of Sines states that, given the triangle below, sin A / a = sin B / b = sin C / c. With this law, we can calculate the side opposite an angle or the angle opposite a side given an opposite angle and side. In the case at right, we know that the side opposite the 70° angle is 6 units long, meaning that the sin angle / opposite ratio for this triangle is 0.156615. Therefore, sin 60° / b = 0.156615, sin 60° / 0.156615 = b, and b = 5.52963. We can also calculate that angle C = 50° (all angles in a triangle add to 180°) and through a similar calculation to the above, find out that c = 4.89124. Note: All triangles do not have a ratio of 0.156615. Each triangle has a different sin angle / opposite ratio. Figure 1
The Law of Cosines
The Law of Cosines is another law to help solve triangles. It states that c2 = a2 + b2 - 2ab cos C. This law enables us to calculate the measure of any angle given the three sides, or the third side given an angle and two sides. In the triangle below, we can calculate the measure of angle A because we know the lengths of sides a, b, and c. 112 = 82 + 102 - 2(8)(10)cos C, (121 - 64 - 100) / (-160) = cos C, 0.26875 = cos C,74.4101° = C. This calculation can be repeated for angle B (b2 = a2 + c2 - 2ac cos B) to learn that its measure is 61.1215°. Therefore, angle A = 180° - 74.4101° - 61.1215° = 44.4744°. The triangle is solved.
Inverse Trigonometric Functions
Inverse trig functions, such as Sin-1, return the appropriate angle given the ratio between two sides. For instance, Sin-1 (6 / 7) = 58.9973°. (see figure 3). Inverse trig functions, with the two laws stated above allow us to solve triangles. In fact, Cos-1 was used above to arrive at the last step in the section on the Law of Cosines.
Note: The abbreviations of inverse trig functions are capitalized because the domain of the regular function is restricted so the inverse is a function. Also, the -1 in Sin-1 has nothing to do with multiplicative inverse. It is only a form of notation.
When solving a triangle, it is important to classify it into one of four cases:SSS, ASA, SAS, or AAS. (The other possibilities, AAA and SSA, do not define triangles.) In the SSS case, the Law of Cosines and Cos-1 is used, as above, to find the first angle. Other angles can be calculated likewise or with the Law of Sines. The ASA case first uses the fact that all angles of a triangle add to 180° and then the Law of Sines is used to find the remaining two sides. The SAS case requires the Law of Cosines and then the Law of Sines. The AAS case can be solved the same way as the ASA case. However, the SSA case does not work so well. Using the Law of Cosines, we get a quadratic yielding two answers, and using the Law of Sines, we must use Sin-1, which also yields two answers. (Cos-1 does not.) These problems only occur if the angle is less than 90°, but the SSA case does not always define a triangle. The AAA case is somewhat unique. It defines the ratio of the sides to each other, but the measure of one side must be known to calculate the others. In fact, AAA proves similarity between triangle, but cannot prove congruency.