Solving Real World Problems Using the Law of Sines

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  • 0:01 Law of Sines
  • 1:36 Real World Problems
  • 2:23 Example 1
  • 3:52 Example 2
  • 5:53 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will be able to use the law of sines to solve problems that you will encounter in the real world. Learn what kind of shape you need in order to solve your real world problems.

Law of Sines

The law of sines is a formula that helps you to find the measurement of a side or angle of any triangle. As you know, our basic trig functions of cosine, sine, and tangent can be used to solve problems involving right triangles. But what about other triangles? That's where the law of sines comes in. This trigonometric law lets you solve problems involving any kind of triangle that you come across. As long as your shape is a triangle, you can use the law of sines to help you solve the problem. Let's take a look at this formula. It looks like this:

law of sines

The small letters a, b, and c stand for the sides of our triangle, and the big letters A, B, and C stand for the corresponding opposite angles to these sides. So, angle A is opposite side a, angle B is opposite side b, and angle C is opposite side c.

law of sines

Look carefully at the formula. Do you see that it actually has three parts? It has two equal signs separating them. Because of this, we get to choose which two parts we are going to use. If we are dealing with just sides a and b and their corresponding opposite angles, then we need only the first two parts. If we are dealing with sides a and c and their corresponding angles, then we need only the first and third parts.

Also looking at the formula, we see that if we are looking for a missing side, then we need to know the opposite angle of the missing side along with another side and its opposite angle. If we are looking for a missing angle, then we need to know the measurement of the opposite side along with another angle and its opposite side.

Real World Problems

What kinds of problems can this useful trigonometric law help us solve? Any problem that involves triangles. Where do triangles come into play in the real world? Well, think of a tower that is leaning. Now picture yourself standing a distance away from the tower. Someone comes up to you and frantically asks for your help. There is someone stuck on the third floor of the leaning tower. They need to know how long of a ladder is needed so they can reach the third floor if they place the ladder where you are standing. If you drew this problem out on paper, you would see that you get a triangle.

law of sines

Because you have a triangle, you can use the law of sines to help you solve it. You will find problems like this and other triangle problems in the real world. Do you want to see how we can solve this leaning tower problem?

Example 1

Let's do it:

You are standing 10 feet away from a certain leaning tower. Find the ladder length that is needed to reach the third floor of the leaning tower if the ladder is placed where you are standing. The angle that is formed by the tower and the ground is 108 degrees, and the angle formed by the tower at the third floor and the ladder will be 20 degrees.

We begin by first drawing out our problem. We draw our triangle and label it with the information that is given.

law of sines

Okay. We see that the side we want to find is the ladder side. We can arbitrarily label that with a. Because we are dealing with any kind of triangle, it doesn't matter which side we label with a, b, or c. If we label the ladder side as a, then our angle A is 108, the angle opposite our side a. We can go ahead and label the 10 as side b. Our angle B is then 20.

Looking at what we have now, we see that we only need the first two parts of our law of sines to find our answer: a/sin A = b/sin B. Plugging in our values, we get a/sin 108 = 10/sin 20. Using algebra to help us solve for a, we get a = (10/sin 20)*sin 108 = 27.8. We have found our answer. The ladder length that is needed is 27.8 feet.

Example 2

Let's try another example:

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