Law of sines, Trigonometry of Triangles

Formula to find angle and side lengths

The law of sines provides a formula that relates the sides with the angles of a triangle. This formula allows you to relatively easily find the side length or the angle of any triangle.

Law of Sines Formula

formula and picture of the law of sines

When to use the law of sines formula

You should use the law of sines when you know 2 sides and an angle (case 1 in the picture below) and you want to find the measure of an angle opposite a known side. Or when you know 2 angles and 1 side and want to get the side opposite a known angle (case 2 in picture below). In both cases, you must already know a side and an angle that are opposite of each other.

when to use law of sines formula

Cases when you can not use the Law of Sines

The picture below illustrates a case not suited for the law of sines. Since we do not know an opposite side and angle, we cannot employ the law of sines formula.
By the way, we could use the law of cosines to find the length of the side opposite the 115° angle. Once we know that side length ,we'd be able to use the law of sines as indicated by case 1 in the picture up above.


on how to use the Law of Sines

Practice Problems

Problem 1

Use the formula for law of sines to determine the measure of angle b to the nearest tenth

Step 1

set up ratio with known pair of side/angles

$$ \frac{ sin( \color{red}{ b})}{ 16} = \frac{ sin(115)} {123} $$

Step 2

solve for unknown

$$ \frac{ sin( \color{red}{ b })}{ 16} = \frac{ sin(115)} {123} \\ sin( \color{red}{ b }) = \frac{ 16 \cdot sin(115)} {123} \\ sin( \color{red}{ b }) = 0.11789369587468619 \\ \color{red}{ b } = sin^{-1} ( 0.11789369587468619 ) \\ \color{red}{ b } = 6.770557323410266 \\ \color{red}{ b } \approx 6.8 $$

Problem 2

Is it possible to use the law of sines to calculate x pictured in the triangle below?

picture of law of sines problem
Step 1

Yes, first you must remember that the sum of the interior angles of a triangle is 180 in order to calculate the measure of the angle opposite of the side of length 19.
Now that we have the measure of that angle, use the law of sines to find value of x

Step 2

solve for unknown

Set up the equation:

$$ \frac { \color{red}{x} }{ sin(116)} = \frac{19}{sin (34) } $$

then cross multiply to calculate x.
x = 30.5

When there are no Triangles!

So if you are given any 2 sides and an angle, can you always use the law of sines to find all the other sides and angles of a triangle?

The answer is Not always!

Consider the following. Let's imagine that we know that there is some 'triangle' ABC with the following information:
BC = 23 ,AC = 3, = 44.
Below represents an accurate picture of these sides and angles. Red represents a given of the triangle (2 sides and 1 angle). Grey represents a variable that we can change (the grey 'side' AB). The gray line represents the third side of the triangle that we don't know. The only thing we know about that side is that it must extend at 44 degrees. As you can see from the picture, there is no way for that grey side to rise at 44 degrees and also to intersect with point A. Therefore no triangle can be drawn with these givens.

Impossible 'Triangle'

Law of Sines, No Triangle

Side BC = 23
Side AC = 3
$$ \angle B$$= 44

Let's see what happens, if we try to use the law of sines to determine the length of "side AB,", the grey line in the picture on the left.

You know the deal, lets use our law of sines formula

$$ \frac{sin(44) }{3 } = \frac{sin(a)}{23} \\ sin(a) = \frac{23 \cdot sin(44)}{3} = 5.32571417 \\ a = sin^{-1}( \color{red}{ 5.32571417}) $$

As you can see angle A is 'impossible' because the sine of an angle cannot be equal to 5.3. (Remember the greatest value that the sine of an angle can have is 1)

Double triangle Problems

Problem 3

What is the length of side CB?

Step 1

Use the fact the sum of the interior angles of a triangle is 180° to calculate all of the angles inside the triangles.

Step 2

Now, use the law of sines formula to set up an equation.

Step 3

From here you have several options

Step 4

The easiest one is to use sohcahtoa to solve for a side length to arrive at the answer.

next toLaw of Sines vs Cosines