# Law of sines, Trigonometry of Triangles

Formula to find angle and side lengths

### Law of Sines Formula

The law of sines formula allows us to set up a proportion of opposite side/angles (ok, well actually you're taking the sine of an angle and its opposite side).

For instance, let's look at Diagram 1.

One side of the proportion has side A and the sine of its opposite angle

The other side of the proportion has side B and the sine of its opposite angle

#### When to use the law of sines formula

What You Know What You Can Find
I side and its opposite angle
II. side
angle opposite kown side

What You Know What You Can Find
I side and its opposite angle
II. angle
side opposite kown angle

#### In a nutshell, you need one opposite angle/side pair and one other measurement

There are two different cases when you use this formula. But really, there is just one case .

To use the law of sines, you need to know one opposite angle/side pair measurements. If you have that one pair, then all that you need is a single measure of anything else (a side , an angle) and you can solve for that thing's opposite.

#### 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.

Below is another example of a problem that could not directly solved by our law of sines formula because we do not know 1 side of opposite angle sides.

*DPF caram *

### Practice Problems

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

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

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?

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'

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

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.

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