Write the equations given in the Law of Sines and the Law of Cosines. Recall that solving a triangle means to find all the unknown measures of angles and sides.
See solution.
Practice makes perfect
We start by recalling that solving a triangle means to find all the unknown measures of angles and sides. Next, let's write the equations given by the Law of Sines and the Law of Cosines.
Law of Sines
Law of Cosines
sin A/a = sin B/b = sin C/c
a^2 = b^2+c^2 - 2bccos A b^2 = a^2+c^2 - 2accos B c^2 = a^2+b^2 - 2abcos C
Let's consider the triangle shown below.
Depending on how much information we know about a triangle, we can use either the Law of Sines or the Law of Cosines.
Knowing Two Angles and One Non-Included Side
If we know the measures of two angles and the length of one non-included side (AAS), we solve the triangle by using the Law of Sines. In the diagram below, we show how to solve the triangle if we know m ∠ A, m ∠ B, and a.
Knowing Two Angles and the Included Side
If we know the measures of two angles and the length of the included side (ASA), we also use the Law of Sines. In the following diagram, we show how to solve the triangle if we know m ∠ A, m ∠ B, and c.
Knowing Two Sides and One Non-Included Angle
If we know the length of two sides and the measure of one of the non-included angles (SSA), we can use the Law of Sines to solve the triangle. First, we find the sine of the second non-included angle.
By taking the inverse sine, we find m∠ B. Notice that at this time, we know two angles and two sides, then we can proceed as in the first case studied above.
Knowing Two Sides and the Included Angle
If we know the length of two sides and the measure of the included angle (SAS), we can solve the triangle by using the Law of Cosines. First, we find the length of the third side.
Next, we solve the other two equations for the cosine of the corresponding angles.
cos A = b^2+c^2 - a^2/2bc [0.2cm]
cos B = a^2+c^2 - b^2/2ac
By taking the inverse cosine, we find the measure of the missing two angles.
Knowing The Three Sides
If we know the length of the three sides of a triangle (SSS), we can solve it by using the Law of Cosines. To do that, we proceed as the final part of the previous case. That is, first we solve each equation for the cosine of the angle and then take the inverse cosine to find the angle measure.
Conclusion
From what we've seen above, we can make a table where we write which method to use to solve a triangle depending on the given parts of it.
