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

formula and picture of the law of sines

When to use the law of sines formula

In a nutshell, you use the Law of Sines, when you know 2 angles and the non-included side or when you know two sides and the non-included angle.

What You Know What You Can Find
2 sides and the non-included angle angle opposite kown side

Law Of sines case I

We can use the formula because we are working with 2 opposite pairs of sides/angles .

What You Know What You Can Find
2 angles and the non-included side side opposite kown angle

Law Of sines case II

We can use the formula because we are working with 2 opposite pairs of sides/angles .


Two different Cases? or Really just one?

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

Just look at it You can always immediately look at a triangle and tell whether or not you can use the Law of Sines-- you need 3 measurements : either 2 sides and the non-included angle or 2 angles and the non-included side.

It's all about opposites To use the law of sines, you need to know one opposite angle/side pair measurements.

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 an opposite side and angle, we cannot employ the formula.

Non example for Law of Sines
Or, just look at it: Remember when you have 2 sides, the angle must be non-included.
By the way, we could use the law of cosines to find the length of the side opposite the 115° angle.

Do you know when to use the formula?

In which triangle(s) below, can we use the formula?

Both triangles below have 3 known measurements.

Triangle 1 has only one opposite pair that we are dealing with, but that does not help us because we need to know both the angle and itsopposite side.
Or, just look at it: Remember when you have 2 sides, the angle must be non-included.


Triangle 2 does have 2 opposite pairs that we are dealing with, and we do know the measurements of one angle and its opposite side.
Or, just look at it: We can use the formula, when we have 2 sides and the non-included angle.

Identify when to use the law of sines

In the full practice problems below, we will solve for triangle 2's unknown angle measure.

Video

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 proportion with 2 pairs of opposite side/sine of angles

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

opposite sides laballed law of sines
Step 2

solve for unknown

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

Problem 2

Can we use the law of sines to solve for the labelled angle?

opposite sides laballed law of sines

Answer

No, because we need to know the measure of 1 opposite side and angle .
We can not use side with length 20 because we don't know its opposite angle
And we can't use 66 ° angle because we don't know its opposite side.
And, of course, we do not know the measure of the angle opposite of the side of length 13 because...well, because that's the very thing we are solving for!

Just look at it Remember when you have 2 sides, the angle must be non-included.

Problem 3

Can we use the law of sines to solve for the labelled side?

opposite sides laballed law of sines

Answer

yes, because
we need to know the measures of one opposite side and angle which we have with the 29 ° angle and the side of length 11 .

and we know the angle (118 °) opposite the side length that we are solving for opposite sides laballed law of sines

Or just look at it Remember when you have 2 angles, the side must be non-included.

Problem 4

(follow up from question 3). Now, use the formula for law of sines to determine the measure of the laballed side to the nearest tenth

opposite sides laballed law of sines
Step 1

set up proportion with 2 pairs of opposite side/sine of angles

$$ \frac{ \red b}{ sin(118)} = \frac{ 11 } {sin(29)} $$

opposite sides laballed law of sines
Step 2

solve for unknown

$$ \frac{ \red b}{ sin(118)} = \frac{ 11 } {sin(29)} \\ \red b = \frac { sin(118) \cdot 11 }{ sin(29)} \\ \red b = 20.033479361472256 \\ \red b \approx 20.0 $$

Problem 5

For $$ \triangle DEF $$, find $$e$$ to the nearest hundredth.

triangle for law of sines

(Question taken from our law of sines downloadable pdf worksheeet)

Step 1

set up proportion with 2 pairs of opposite side/sine of angles

$$ \frac{ \red e}{ sin(67)} = \frac{ 7 } {sin(54)} $$

triangle for law of sines
Step 2

solve for unknown

$$ \frac{ \red e}{ sin(67)} = \frac{ 7 } {sin(54)} \\ \red e = \frac { sin(67) \cdot 7 }{ sin(54)} \\ \red b = 20.033479361472256 \\ \red b \approx 7.96 $$

Challenge Problems

Challenge Problem 1

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 the 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

Challenge Problem 2

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

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