The Law of Sines
Home > Lessons > Law of Sines
Search | Updated March 18th, 2017
Introduction

   Triangles are a fundamental geometrical shape. Triangles appear within several disciplines, some of which are architecture, engineering, astronomy, and chemistry. This is why mathematicians have studied them and consequently have several relations to enumerate their sides and angles. Here is one such relation.

   This is the equation called The Law of Sines.

The Law of Sines Equation

   As it will be thoroughly explained within the next section, this equation is useful when a certain angle and its opposite side are known (and either another side or angle) for a given triangle. If this given information is known, then it is possible to calculate another known angle's opposite side or another known side's opposite angle.

   If you are not familiar with how to label a triangle or you have never worked with sine, cosine, or tangent, please read our trigonometry basics section before moving on to the next sections within this page.


   When solving for unknown sides and angles of a triangle, we must first determine how many triangles exist. Most of the time there is only one possible triangle but there is a special set of circumstances that leads to several options.

   If we are ever given two side lengths and the angle that is not between the sides, we have to be careful. This case is called Side-Side-Angle (abbreviated as SSA). Depending on the circumstances, there are three cases: a) no triangle can be made, b) one triangle can be made, or c) two triangles can be made.

   This helpful table can be used to assist us as we determine how many triangles exist.

The Law of Sines: Ambiguous Case (SSA)

   This video will inform you how to use the table and determine how many triangles exist given a SSA situation.

    ideo: The Law of Sines: Ambiguous Case (SSA)
    uiz: Law of Sines: Ambiguous Case (SSA)


   Let's say we are given the following information for a triangle.
Given Triangle

   This situation is appropriate for The Law of Sines. The reason it is appropriate is because the 15 degrees and the 7.5 miles are opposite of each other. This is emphasized in the next graphic.
Angle and its opposite side are known

   Consequently, we will be able to solve for the side marked with the 'x.' [The angles have been marked with the letters 'A,' 'B,' and 'C.' The labeling is arbitrary and the problem could be solved with any arrangement of the letters.]
Problem proposed

   The reason side-x can be calculated is because it is opposite a known angle, angle-C.
Unknown side is opposite known angle

   Now we will refer to our equation. The Law of Sines has three ratios — three angles and three sides.

The Law of Sines

   We have only three pieces of information. We also know nothing about angle-A and nothing about side-a. So, we will only need to utilize part of our equation, which are the ratios associated with 'B' and 'C.'

The Law of Sines for B and C only

   We know angle-B is 15 and side-b is 7.5. We also know that angle-C is 20, but we do not know side-c. We will substitute these values into the equation as such.

Substitude known values

   Now that we were able to construct two equal ratios, called a proportion, let's solve it by cross-multiplying.

Proportion

   Cross-multiplying will give us these equal products.

Products

   The equal products above make a simple equation. We have to solve for the variable 'c.' To do so, we will have to divide both sides of the equation by the sin 15.

Dividing by sin 15

   We can cancel the sin 15s because we're both multiplying and dividing 'c' by that value.

Canceling the sin 15

   With the sin 15 out of the way, we need to plug this into a calculator.

Calculating for c

   After evaluating with a calculator, this is our final value for 'c.'

The value of c

   This means the value of 'x' in our problem is 'c' also, which is 9.9 miles.

   Use this video and quiz to reinforce the lesson.

    ideo: The Law of Sines: Find a Side
    uiz: Law of Sines: Find a Side


   For our next Law of Sines problem, let's take a look at this given information.
Given information

   This stuation is ideal for The Law of Sines. We were given a side and it's opposite angle.
Given information

   Since we also have another side, we can solve for another angle, which has been marked with a 'y' in this graphic.
Solve for unknown angle

   Angle-y rests opposite the 250, which is a known length of our triangle.
Demonstrate opposite side and angle

   When there are two pairs of opposite sides and angle, we can utilize The Law of Sines. All we have to do now is label the triangle. The triangle below has been arbitrarily labeled with 'A,' 'B,' and 'C.' Any arrangement of 'A,' 'B,' and 'C' can be used; so, we'll stick with this arrangement of letters and angles.
Labeling the triangle

   This is when we use The Law of Sines.

The Law of Sines

   We can see that Angle-C is 48 and side-c is 200. Angle-B is unknown, but side-b is 250. So, we will place these values into the equation. Without knowing anything about side-a and angle-A, we'll use only the ratios that involve 'B' and 'C.'

The Law of Sines for B and C only

   This is what the equation looks like once the appropriate numbers are plugged inside of it.

Plugging in values

   When we have two equal ratios, called a proportion, we need to cross-multiply to advance.

Cross multiply to solve

   This is what the cross-products look like.

Cross multiply to solve

   To continue solving for B, we have to divide both sides of the equation by 200.

Divide both sides by 200

   Since we are multiplying and dividing by 200, we can cancel it to get sinB alone.

Canceling the 200

   This is what we are left with, the sin of angle-B.

sinB

   To cancel the sine of angle-B, we have to utilize the inverse sine function.

inverse-sinB

   Therefore, this is what we need to plug into a calculator.

Calculate for B

   This is our final answer for angle-B.

The value B

   Since we were trying to solve for variable-y, it is also 68.3 degrees.

   Use this video and quiz to reinforce the lesson.

    ideo: The Law of Sines: Find an Angle
    uiz: Law of Sines: Find an Angle


   There are two laws, The Law of Sines and the Law of Cosines (see the Related Lessons section for the lesson). This table will help determine which law to use given the circumstances that are present.

Choosing the Correct Law: Law of Sines or Law of Cosines

   Let it be known to use the Law of Sines we do have to know three pieces of information: 1) we either know two angles and a side or 2) two sides and an angle.

   When considering the first case, knowing two angles of a triangle means we can find the third angle. If we know a side, we automatically know the angle opposite that side because we know all the angles of the triangle. For the second case, we must know an angle that is opposite one of the two given sides. Having an angle and a side opposite it is the indicator we must use the Law of Sines.

   Try out our videos on this topic.

    ideo: The Law of Sines: Ambiguous Case (SSA)
    ideo: The Law of Sines: Find a Side
    ideo: The Law of Sines: Find an Angle

   Try out our quizmaster on this topic.

    uiz: Law of Sines: Ambiguous Case (SSA)
    uiz: The Law of Sines: Find a Side
    uiz: The Law of Sines: Find an Angle

   Try out our quizmaster on this topic.

    ctivity: The Law of Sines

   We would like to invite you to learn from our other advanced trigonometry lesson.

    esson: Trigonometric Angles
    esson: The Law of Cosines
    esson: Trigonometric Expressions