Amy has a master's degree in secondary education and has taught math at a public charter high school.
Law of Sines
There are many useful formulas in trigonometry. The one we will be talking about in this video lesson is the law of sines, which tells you how the sides and angles of a triangle are related. By using this formula, you can solve any triangle for a missing side or angle. Recall that your basic trig functions of sine, cosine, and tangent can easily be found by using a right triangle. Remember SOHCAHTOA? It tells you that:
- Sine is equal to opposite/hypotenuse
- Cosine is equal to adjacent/hypotenuse
- Tangent is equal to opposite/adjacent
While our basic trig functions are fractions of the various sides of a right triangle, by using our trig functions in a more elaborate way, we can actually use them to solve problems related to any kind of triangle, not just right triangles. This opens up a whole new world of problems that we can solve using trigonometry. Let's take a look at the law of sines. It looks like this:
The small lower case letters refer to the sides of a triangle. The larger upper case letters refer to the corresponding angles of the triangle. So, angle A is opposite side a, angle B is opposite side b, and angle C is opposite side c.
Solving a Triangle
By using this formula, we can solve any kind of triangle for a missing side or missing angle. Look carefully at the formula and you will see that there are two equal signs. This means that we can use any two parts of this formula; we don't have to use all three parts. The formula does tell us that all three parts are equal to each other. So, what information do we need to find a missing side?
Well, let's look at what happens when we use just two parts of the formula. Say we have a/sin A = c/sin C. If we want to find side c, then we need to know all the other letters, we need to know side a, angle A and angle C. What does this tell us? It tells us that if we are looking for a particular side, we need to know the measurement of one side along with its opposite angle and then we need to know the measurement of the angle opposite the side we are looking for.
What about finding a missing angle? What information do we need to know for that? Let's look. Say we are using this formula - a/sin A = b/sin B. We are looking for angle A. What information do we need in order to solve this formula? We need to know sides a and b along with angle B. What does this tell us? It tells us that in order to find a missing angle, we need to know the measurement of the side opposite the angle we are looking for and the measurement of another side along with its opposite angle.
If we are given a triangle problem to solve, we need to look at what information is given to us and then we can decide which two parts of the law of sines we are going to use. For example, if we are looking for side c and we are given angle C, side b, and angle B, then we will use the formula b/sin B = c/sin C because this contains all the information I have been given and want to find.
Let's look at a couple of examples to see how this is done.
Find side b.
In this problem, what are we looking for? We are looking for side b. Okay, since we are looking for a missing side, that means that we need to know angle B and another measurement of a different side along with its opposite angle. Okay. What are we given?
We are given angle B as 105. So far so good. Now we need another side. Look, we have side a which is 7. We also have angle A which is 35. Great! We have everything we need to find our missing side. Since we are using sides a and b, we will use the two parts of our law of sines that include those sides. We have a/sin A = b/sin B. We can plug in the numbers that we know.
We get 7/sin 35 = b/sin 105. To solve, we use our awesome algebra skills. We also remember to set our calculator to degrees since we are working with degrees. We get b = (7 sin 105)/sin 35 = 11.79. Our side b measures 11.79. We are done. That wasn't too bad, was it?
Find angle A.
Here, we see we are given two sides, one angle, and we need to find another angle. So, let's see, we are given side b along with angle B and we are given side a. We want to find angle A. So we can use the formula a/sin A = b/sin B.
Plugging in the information that we have, we get this - 4.7/sin A = 5.5/sin 63. Again, using our awesome algebra skills to solve for our missing angle A, we get A = sin^-1 ((4.7 sin 63)/5.5). Evaluating this, we get angle A = 49.58 degrees. Remember, since we are looking for an angle, we need to take the inverse sine of our calculations to find our angle in degrees. Also remember, since we are working with degrees, we need to make sure our calculator is also set to degrees.
What have we learned? We've learned that the law of sines tells us how the sides and angles of a triangle are related. It also allows us to solve any triangle for a missing side or angle. The formula looks like this:
The lower case letters stand for the sides of our triangle and the upper case letters stand for the corresponding opposite angles of those sides. This formula has three equal parts to it. To use it, we only need to use two of the parts. We use the parts that contain the information we have and the information that we need to find.
In order to find a missing side, we need to know the measurement of the angle opposite our missing side along with the measurement of another angle with its opposite side. In order to find a missing angle, we need to know the measurement of its opposite side along with the measurement of another side with its opposite angle.
When we have all the information we need, we plug in the values into our formula of two parts. We then use our algebra skills to solve for our unknown. If we are finding an angle, we remember to use the inverse sine operation at the end to find the measurement of our angle.
After this lesson is finished, you should be able to:
- State the meaning of SOHCAHTOA
- Recall the law of sines
- Calculate the missing side or angle of a triangle using the law of sines
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