The Law of Sines Study Guide
The Law of Sines
In this lesson, we prove a powerful theorem called the Law of Sines. This enables us to find the angles and sides of non-right triangles. With the trigonometric functions, we can measure the sides and angles of right triangles. To study a non-right triangle, we break it into right triangles. The first major result in this area is called the Law of Sines.
To begin, take any triangle and suppose the lengths of its sides are A, B, and C. Suppose the measure of the angle opposite side A is a, the angle opposite B is b, and the angle opposite C is c. This is illustrated in Figure 17.1.
If the proof on the next two pages is difficult to follow, you may skip straight to the Law of Sines on page 164.
We can draw a line from the vertex at angle c that makes a right angle with side C. The length of this line is called the height h of the triangle (see Figure 17.2).
This new line creates two right triangles, shown separately in Figure 17.3.
The one on the left has an angle of measure b. Opposite this angle is a side oflength h. The hypotenuse has length A. Thus,
The other right triangle has angle a. Here,
If we put the two of these equations together, we get
B · sin(a) = h = A · sin(b)
If we divide both sides by AB, we get part of the Law of Sines:
The full Law of Sines is
proving that is done by the same method as above. It can be a little bit tricky, however, if one of the angles is obtuse (greater than 90°), as is the case with angle c in our example. First, we rotate the triangle so that side A is across the bottom. When we try to make a right angle from the vertex of angle a to the side A, we must extend side A and have the right angle outside of the triangle, as is illustrated in Figure 17.4.
The two right triangles that are formed have angles of measure b and 180° – c, as shown separately in Figure 17.5.
Thus, we get
We will see in just a moment that sin(180° – c) = sin(c); thus, C · sin(b) = h = B · sin(c), so , as desired. The proof comes from looking at the unit circle in Figure 17.6.
The point on the unit circle that corresponds to the angle θ has the same height (sine) as the point that corresponds to the point 180° – θ. This is why sin(l80° – θ) = sin(θ). Thus, we have completely proven the Law of Sines:
Sometimes the Law of Sines is given instead as the equivalent or, equivalently:
This can be used to find the length of a side of a triangle, provided that we know the measure of one side and two angles.
Find the length x of the side in Figure 17.7.
The length x is opposite the 120° angle, and the 10-foot side is opposite the 35° angle. Thus, we can use the law of sines:
The only important thing is that the angle and side in each fraction are opposite each other. With cross multiplication, we get
Thus, the length of this side is about 15.1 feet long.
Find the length x in Figure 17.8.
We can't immediately use the Law of Sines because we don't know the measure of the angle opposite the 30-inch side. However, this can be easily calculated because the angles of a triangle must add up to 180°. The angle opposite the 30-inch side measures 180° – 48° – 71° = 61°. With this, we can now use the Law of Sines:
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