Section 4.6 The Law of Sines and the Law of Cosines
Live your life as though your every act were to become a universal law. — Immanuel Kant, German philosopher, 1724 – 1804
Problem: Evaluate \(x\text{.}\) See Figure 4.61.
Notation: Vertices, Sides, and Angles of a Triangle See Figure 4.62.
In a triangle \(\triangle ABC\) with vertices \(A\text{,}\) \(B\text{,}\) and \(C\text{:}\)
\(\alpha\) is the angle at \(A\text{.}\)
\(a\) is the side opposite \(A\) and \(\alpha\text{.}\)
\(\beta\) is the angle at \(B\text{.}\)
\(b\) is the side opposite \(B\) and \(\beta\text{.}\)
\(\gamma\) is the angle at \(C\text{.}\)
\(c\) is the side opposite \(C\) and \(\gamma\text{.}\)
The Law of Sines.
We use Figure 4.63.
Observe that
Now we calculate the length of \(h\) in two different ways:
Hence
Similarly, \(c \sin \alpha = a \sin \gamma\text{.}\)
We write
Since \(b \sin \alpha = a \sin \beta\) and \(c \sin \alpha = a \sin \gamma\text{,}\) it follows that
This is known as the Law of Sines.
Example 4.6.1. Law of sines.
In the triangle \(\triangle ABC\) with the usual notation, suppose that \(\sin \alpha = \frac{2\sqrt{2}}{3}\text{,}\) that \(a = 11\) and that \(b = 9\text{.}\) Find \(\sin \beta\text{.}\)
By the Law of Sines
Area of a Triangle: The Law of Sines implies that
See Figure 4.64.
In \(\triangle ABC\) with the usual notation, let \(h\) be the hight to the side \(c\text{.}\)
Observe that \(h=a\sin \beta\text{.}\) Thus, the area of the triangle is given by
Note: Notice that in this formula, \(\beta\) represents the anglebetween \(a\) and \(c\text{.}\)
Example 4.6.2. Use the given area to find a side.
In \(\triangle ABC\) with the usual notation, suppose that \(\sin \alpha = 2/3\text{,}\) that \(b = 9\) and that the area of \(\triangle ABC\) is \(24\) Find \(c\text{.}\)
Note that, in the usual notation, \(\alpha\) represents the angle between \(b\) and \(c\text{,}\) so the area of \(\triangle ABC\) is given by
We conclude that in the given triangle
The Law of Cosines. See Figure 4.65.
See Figure 4.65.
We use Figure 4.65.
Observe that, by the Pythagorean theorem,
The fact that
is known as the Law of Cosines.
Note: Observe that the the angle \(\beta\) is between the sides \(a\) and \(c\) and that side \(b\) is opposite to the angle \(\beta\text{.}\) Hence
Example 4.6.3. Solve using the Law of Cosines.
Evaluate \(x\text{.}\) See Figure 4.66.
By the Law of Cosines,
We solve the quadratic equation to obtain
and, since \(x\gt 0\) (why?), we reject the negative option.
Thus