The Law of Sines Study Guide (page 2)
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:
Finding Angles with the Law of Sines
We can use the Law of Sines to find any side of a triangle if we know two angles and one side. If we try to find the measure of an angle given two sides and one angle, however, a problem arises. For example, suppose we have the triangle in Figure 17.21. What is the measure of angle x?
Because we have two pairs of opposite angles and sides, we can set up the Law of Sines:
Our instinct is to evaluate x with the inverse sine function:
If we know that Figure 17.21 is drawn accurately, then this is the measure of angle x. However, if Figure 17.21 is not drawn accurately, then it is possible that the triangle is really the one in Figure 17.22.
In this case, x is an obtuse angle (greater than 90°) and thus could not be 70°.
In Figure 17.22, the measure of angle x is 110°, the supplement of 70°. This is because sin(110°) = sin(70°) as discussed after Figure 17.6:
sin(180° – θ) = sin(θ)
The inverse sine function will output angles only between –90° and 90°. That is to say, it only outputs acute angles. If we are seeking the measure of an acute angle θ with sin(θ) = r for a specific positive ratio r, then θ= sin–1(r). If we want the measure of an obtuse angle θ with sin(θ) = r, then θ = 180° – sin –1(r).
Find the measure of the obtuse angle x in Figure 17.23.
We have x opposite the side of length 10 inches and 32° opposite the side of length 5.6 inches. Thus, we can use the Law of Sines:
Using the inverse sine function, we get
sin–1(0.9463) ≈ 71.1°
This cannot be the measure of angle x because x is obtuse. Thus, x must be the supplement of this angle.
x ≈ 180° – 71.1° =108.9°
Suppose the triangle in Figure 17.24 is drawn accurately. Find the measure of angle x.
Here, we use the Law of Sines:
sin–1(0.8631) ≈ 59.7°
Because x is an acute angle, this measure is correct: x ≈ 59.7°. If x had been obtuse, then its measure would be 180° – 59.7° = 120.3°.
Practice problems for this study guide can be found at:
Today on Education.com
- Coats and Car Seats: A Lethal Combination?
- Kindergarten Sight Words List
- Child Development Theories
- Signs Your Child Might Have Asperger's Syndrome
- 10 Fun Activities for Children with Autism
- Why is Play Important? Social and Emotional Development, Physical Development, Creative Development
- Social Cognitive Theory
- GED Math Practice Test 1
- The Homework Debate
- First Grade Sight Words List