Review how sine and cosine can help us find the angle measures and side lengths of a right triangle. How can we adapt these relationships for a non-right triangle?
See solution.
Practice makes perfect
Let's consider the right triangle below, for which we know all its angle measures and side lengths.
Next, let's find some particular values and ratios that will help us deduce the required laws.
Law of Sines
Let's find the ratios of the sine of each angle to the length of the corresponding opposite side.
sin A/a
sin B/b
sin C/c
sin 60^(∘)/3.46 = 0.25
sin 30^(∘)/2 = 0.25
sin 90^(∘)/4 = 0.25
As we can see, the three ratios above are equal. The good news is that this is true for any triangle, not only for the right triangles, and this is called the Law of Sines.
Law of Sines
Let a, b, and c be the side lengths of a triangle, and let A, B, and C be the angles opposite to each side, respectively. Then, we have that sin A/a = sin B/b = sin C/c
Law of Cosine
In this part, we start by finding the following values, and then we will try to find a relation between them.
a^2
b^2
c^2
b
c
cos A
12
4
16
2
4
0.5
From the data written in the table above, we can write the following equation.
12 = 4^2 + 16 - 2* 2* 4* 0.5
⇕
a^2 = b^2+c^2 - 2abcos A
As before, the relation written above is true for any triangle. In fact, we can write equivalent equations involving the cosine of the other two angles. These equations are known as the Law of Cosines.
Law of Cosines
Let a, b, and c be the side lengths of a triangle, and let A, B, and C be the angles opposite to each side, respectively. Then, we have that a^2 = b^2+c^2 - 2abcos A
b^2 = a^2+c^2 - 2accos B
c^2 = a^2+b^2 - 2abcos C
