Law of Sines: Exercises & Formula |

To work with triangles, you often need their angles and side lengths. But what if you only gave a few values ​​and you need the others? In such cases, the law of sine can help you.

Law of Sines formula

You can use the law of sines to determine sides and angles in any triangle, as long as you only know one «side-angle pair» and one other quantity.

Figure 1: Law of Sines in a triangle

The three sides and their opposite angles are shown on this triangle. They are each marked in the same color.

the law of sine formula as follows:

The ratio of one side to the sine of the opposite angle is the same as the ratio of all other sides to the sine of their opposite angles. This formula is the basis of the law of sines.

Calculate the law of sine

In school mathematics, you will mostly come across arithmetic problems on the subject of the law of sines. Usually some values ​​are already given and you have to calculate the missing sides and angles.

Figure 2: Law of Sines in a triangle

In this example, the side lengths are c and a as well as the angle specified.

Task 1

Calculate the angle using the law of sines!

solution

Step 1:

Since you have given three quantities here, you can already write down the equation:

Step 2:

Now you can change the formula according to the quantity you are looking for:

Step 3:

Now you put in your values ​​and calculate:

Step 4:

You are still missing one step, because the result is only the sine of the angle you are looking for:

To find out the angle, you can use the sin-1(x) function on your calculator. The x corresponds to the value that we just calculated.

change the sine law

In order to calculate with the law of sine, you first have to convert it so that you can solve it for the value you are looking for.

To repeat the rearrangement of fractions, please have a look at the article «Fractions»!

Since you can rearrange the law of sine in many different ways, the form of the law can be very different. The content always remains the same. It’s all about the angle aspect ratios within a triangle:

In the following example you should change to:

You have the option here to rearrange the original equation, or you can use one from the list above.

Here is an example with the third equation:

This is nothing more than the reciprocal of the original equation. As long as you do the reciprocal on both sides of the equation, their ratio doesn’t change.

Now Dunoch must isolate, in which you c bring to the other side:

In this way you could have, for example, rearranged the equation for our example problem.

The law of sine – determine the necessary values

Sometimes arithmetic problems are set in such a way that not all the necessary sizes of the triangle are given directly. For example, an angle that you need to use the law of sines may be missing. In this case you can calculate the missing angle using the sum of the angles in the triangle.

With this theorem you can calculate the missing third angle given two angles.

exercise 2

Figure 3: Law of Sines in a triangle

Calculate the side length a!

solution

Now set up the formula as before:

The problem: You only gave. That’s one value too few to use the law of sine.

This is where the sum of angles comes into play. The angles are given, so you can calculate:

Now the same applies as before and you can a calculate by the law of sines:

Law of Sines Derivation – Right Triangle

The following section explains how you can derive the law of sines.

A good understanding of the sine is a prerequisite for this derivation. If you are unsure, you can read the article Sine, cosine and tangent on a right triangle.

Take a general triangle and divide it into two right triangles by drawing an elevation.

Figure 4: Proof of the law of sines

Now you set up the sine for the two resulting right-angled triangles:

Now you rearrange both and equate the two terms:

You can now convert this to the first part of the law of sines:

For the last part you need the height to c:

Figure 5: Derivation of the law of sines

Again you have two right-angled triangles and you can set up their sine:

Now you isolate each side again and equate the two terms:

When you put that together with your first formula, the following applies:

You get the full law of sines.

Law of Sines Tasks

So that you can consolidate what you have learned, you will find a few arithmetic problems for the law of sines here. You should calculate both angles and the length of missing sides.

task 3

The following triangle is given. Calculate the length of the side b using the law of sine!

Figure 6: Calculation example of the law of sine

Solution:

For the triangle, the angles are given, as is the side length c. So in this triangle:

All you have to do is follow this formula b convert and calculate:

task 4

The following triangle is given. Calculate the angle using the law of sines!

Figure 7: Calculation example sine law

solution

There are two sides in this triangle, but only one angle. Therefore:

To calculate the angle, you first rearrange the formula and solve it as follows:

Now all you have to do is solve the sine:

task 5

The following triangle is given. Calculate all missing sides and angles!

Figure 8: Calculation example of the law of sine

solution

1st step: calculate

First, use the law of sines again to calculate the angle:

2nd step: calculate

To get on from here, you still need the last angle. You can calculate this by using the formula for the sum of the angles in a triangle.

3rd step: calculate c

Now only the last side is missing in the triangle. You can calculate this again normally using the law of sines:

You have now calculated all the missing sides and angles.

Law of Sines – The Most Important

  • Sine formula:
  • The law of sine applies to all triangles.
  • The law of sines can be used to calculate missing quantities in a triangle as long as at least one angle and its opposite side are known.
  • The law of sine can be written in different forms:
  • Given two angles, the third can be calculated by the sum of the angles in the triangle.
  • The Law of Sines is derived using the sine and two heights in any triangle.