# Law of Sines and Cosines, and Areas of Triangles

This section covers:

# Review of Right Triangle Trig

We learned about Right Triangle Trigonometry here, where we could “solve” triangles to find missing pieces (angles or sides).

Here is a review of the basic trigonometric functions, shown with both the SOHCAHTOA and Coordinate System Methods.   Note that the second set of three trig functions are just the reciprocals of the first three; this makes it a little easier!

Remember that the sin (cos, and so on) of an angle is just a number!

We use the Law of Sines and Law of Cosines to “solve” triangles (find missing angles and sides) when we do not have a right triangle (which is called an oblique triangle).  This is a little more complicated, and we have to know which angles and sides we do have to know which Law to use, but it’s not too bad.

Note that the Law of Sines can still be used to solve Right Triangles, using the 90° angle as one of the angles. It just turns out that the sin of 90° is 1; so it turns into a SOH case.

# Law of Sines

The Law of Sines (or Sine Rule) provides a simple way to set up proportions to get other parts of a triangle that isn’t necessarily a right triangle.

We use the Law of Sines when we have the following parts of a triangle, as shown below:   Angle, Angle, Side (AAS), Angle, Side, Angle (ASA), and Side, Side, Angle (SSA).   Note that it won’t work when we only know the Side, Side, Side (SSS) or the Side, Angle, Side (SAS) pieces of a triangle. (Remember that these are “in a row” or adjacent parts of the triangle).

The only problem is that sometimes, with the SSA case, depending on what we know about the other sides and angles of the triangle, the triangle could actually have two different shapes (one acute and one obtuse).  For these case, we have to account for both those shapes (so we’ll basically have two answers for the triangle, or maybe even no triangle can be formed).   This is called the Ambiguous Case, and we’ll discuss it below here.

Once we know the formula for the Law of Sines, we can look at a triangle and see if we have enough information to “solve” it.   “Solving a triangle” means finding any unknown sides and angles for that triangle (there should be six total for each individual triangle).

Note that we usually depict angles in capital letters, and the sides directly across from them in the same letter, but in lower case:

Let’s do some problems; let’s first use the Law of Sines to find the indicated side or angle.

Also remember that if we can’t solve it from what we have, and we have two of the three angles, we can obtain the third angle from Geometry (the sum of angles in a triangle is 180°).

Also note that the triangles aren’t typically drawn to scale, meaning the angles and side measurements don’t exactly match the pictures.   When I draw the triangles, I typically put the A and B angles on the “ground”.

Also notice that if you try to draw the triangles to scale, you’ll see that larger angles are opposite larger sides, and smaller angles are opposite smaller sides.  This could be a way you can check to see if you’re getting the correct answers.

And don’t forget to put your calculator in “DEGREE” mode.

Also, notice that I tend to round the angle measurements to a tenth of a degree, and the side measurements two decimal places (hundredths).  If you want more accurate measurements when you start calculating the other sides and angles, you can use the STO> function in your calculator, or retype the long decimals you may get.   To do this, use “STO>X” after seeing the value you want stored, and then “X” when you want to retrieve that value.

Here are some problems where we need to “solve” the triangle, using the Law of Sines.  Again, solving the triangle means finding all the missing parts, both sides and angles.

## Law of Sines Ambiguous Case (SSA)

When we have the Side-Side-Angle (in a row) case (SSA), we could have one, two, or no triangles formed, and we have to do extra work to determine which situation we have.

In these cases, I always like to draw my triangle with the known angle on the bottom left, so I can see what’s going on.  If the side directly across from this angle is less than the side touching this angle, we will probably have an ambiguous case (or may have no triangle that can be formed).

In this situation, if we get an error message in the calculator when trying to get the other angle using Law of Sines, there is no triangle that can be formed with the numbers given (and that’s the answer).  If we get an answer other than 90°, we will have two triangles that can be formed, and the second angle is 180 minus the angle we just got (one will be acute and one will be obtuse).   We can then solve for two different triangles.  (The given two sides and one angle for the two triangles will be the same).   If we do get 90° for the second angle, we have one right triangle.

Here’s an illustration of this:

Let’s do some problems, so it won’t seem so confusing.  Solve for all possible triangles with the given conditions:

Note:  We can also solve ambiguous case triangles using the Law of Cosines and a graphing calculator here).

Here’s another type of problem you might encounter when learning about the Law of Sines Ambiguous Case:

Problem:

Given a triangle ABC, what values of a would result in making two triangles if A = 40° and b = 10?

Solution:

Let’s first draw the triangle:

In order for there to be two triangles, a must be less than 10, but greater than the height of the triangle, which is 10sin40°, or 6.427.    So 6.427 < a < 10.

# Law of Cosines

The Law of Cosines (or Cosine Rule) again provides a simple way to set up proportions to get other parts of a triangle that isn’t necessarily a right triangle.

We use the Law of Cosines when we have the following parts of a triangle, as shown below:   Side, Angle, Side (SAS), and Side, Side Side (SSS).  It also will work for the Side, Side, Angle (SSA) case, and we will see that here, but the Law of Sines is usually taught with this case, because of the Ambiguous Case.

(Remember that these are “in a row” or adjacent parts of the triangle).

Once we know the formula for the Law of Cosines, we can look at a triangle and see if we have enough information to “solve” it.   “Solving a triangle” means finding any unknown sides and angles for that triangle (there should be six total for each individual triangle).

Again, note that we usually depict angles in capital letters, and the sides directly across from them in the same letter, but in lower case:

Let’s do some problems; let’s first use the Law of Cosines to find the indicated side or angle.

Remember again that the triangles aren’t typically drawn to scale, meaning the angles and side measurements don’t exactly match the pictures.   When I draw the triangles, I typically put the A and B angles on the “ground”.

Again notice that if you try to draw the triangles to scale, you’ll see that larger angles are opposite larger sides, and smaller angles are opposite smaller sides.  This could be a way you can check to see if you’re getting the correct answers.

And don’t forget to put your calculator in “DEGREE” mode.

Now let’s solve for all possible triangles with the given conditions.

Caution: When using the Law of Cosines to solve the whole triangle (all angles and sides), particularly in the case of an obtuse triangle, you have to either finish solving the whole triangle using Law of Cosines (which is typically more difficult), or use the Law of Sines starting with the next smallest angle (the angle across from the smallest side) first.   This is because of the problem with ambiguous cases with triangles.

## Law of Cosines Ambiguous Case (SSA)

Let’s solve an SSA Ambiguous Case Triangle using the Law of Cosines instead of the Law of Sines.    We can do this fairly easily using a graphing calculator; in fact the calculator can actually tell us how many triangles we will get!   We’ll use the same problem that we used earlier.

# Areas of Triangles

In Geometry we learned that we can get the area of triangles quite easily if we know the base of the triangle and the altitude (which is a line that is perpendicular to the base and extends up to the top of the triangle):     $$\text{Area}\,\,=\frac{1}{2}bh$$  or   $$\text{Area}\,\,=\frac{{bh}}{2}$$.

Now that we know trig, we get the area of a triangle without having to know the altitude if we know two sides, and the angle inside the two sides (the Side-Angle-Side or SAS case), or three sides of the triangle (Side-Side-Side, or SSS case).

And remember that many times if you don’t initially have these cases, you can solve the triangle using the Law of Sines or Law of Cosines to get these angles and sides.

Here are the two formulas; note that one uses the sin function, and the other (Heron’s formula) uses the sides only, with one half the sum of the sides being what is called a semiperimeter:

Here are some examples:

# Applications/Word Problems

Here are some examples of applications of the Law of Sines, Law of Cosines, and Area of Triangles.  One big hint on doing these word problems is to try to draw the diagram (if they don’t give you one) as much to scale as possible, so you can see if your answers make sense!   For example, draw the angles as close to the correct angle measurements and sides in the proportion of the numbers they give you.

Problem:

Ali and Brynn are standing 250 yards apart.  Both girls site an airplane with angles of elevation 40° and 45°, respectively.  How far from the plane is Ali?

Solution:

This is a good example how we might use the Law of Sines to get distances that are typically difficult to measure.  This stuff is really used in “real life”!

Problem:

In a parallelogram ABCD, AD = 450 feet, AB = 240 feet, and diagonal AC = 290 feet.  What is the measure of angle BCD?

Solution:

Sorry to spring some heavy Geometry on you, but you’ll see it’s not that bad.  By definition, a parallelogram is a quadrilateral (four-sided figure with straight sides) that has opposite parallel sides, and it turns out that opposite sides are equal.  Parallel means never crossing, like railroad tracks.

For a parallelogram ABCD, we need to draw A-B-C-D in a row (in any direction) around the figure.

So let’s draw the parallelogram and see how we can solve the problem.  Note that we need to know from Geometry that adjacent angles in a parallelogram add up to 180°.  These are called Same Side Interior angles.

Problem:

Three dogs are sitting in a kitchen and waiting to get their dog food.   It turns out that Dog A is 4.5 feet from Dog B, and Dog C is 2.5 feet from Dog A, as shown in the diagram below (note that the three dogs’ positions form a right (90°) angle).

The angles formed at Dogs A and B to the dog food are 40° and 80°, respectively, as shown in the diagram.

How far is Dog C from the dog food?

Solution:

This one’s a little tricky, since we have more than one triangle.  We’ll first work with the triangle where we have enough information, and then use a common side to solve the parts of the triangle we want.

We can start out using the triangle between Dogs A and B and the dog food, where we have an ASA case (Law of Sines):

Solution:

(Note that this problem is actually solved more easily using vectors here in the Introduction to Vectors section.)

Let’s first talk about what it means to have a bearing of a certain degree, since this is typically used in navigation.  First of all, think of north as going up (positive y axis), south as going down (negative y axis), east as going to the right (positive x axis), and west as going to the left (negative x axis).

Unless otherwise noted, bearing is the measure of the clockwise angle that starts due north or on the positive y axis (initial side) and terminates a certain number of degrees (terminal side) from that due north starting place.   (This is also written, as in the case of a bearing of 40° as “40° east of north”, or “N40°E”).

Note:  Sometimes, you’ll see a bearing that includes more directions, such as 70° west of north, also written as N70°W.   In this case, the angle will start due north (straight up, or on the positive y axis) and go counterclockwise 70° (because it’s going west, or to the left, instead of east).  Similarly, a bearing of 50° south of east, or E50°S, would be an angle that starts due east (on the positive x axis) and go clockwise 50° clockwise (towards the south, or down).   Also, if you see a bearing of southwest, for example, the angle would be 45° south of west, or 225° clockwise from north, and so on.

And each time a boat or ship changes course, you have to draw another line to the north to map its new bearing.

Here are some bearing examples:

In this problem, we also have to remember that Distance = rate x time, since we are given rates and times and need to calculate distances.  We also need to use some Geometry, so if you haven’t had it yet, you may want to skip this problem.

Problem:

A cruise ship travels at a bearing of 45° at 15 mph for 3 hours, and changes course to a bearing of 120°.  It then travels 10 mph for 2 hours.   Find the distance the ship is from its original position and also its bearing from the original position.

Here’s one more bearing problem:

Problem:

Joa is standing 100 feet from her friend Rachel.   From Joa, the bearing to Rachel is E25°S.  From Joa to another friend, Emily, the bearing is S30°E.  The bearing from Rachel to Emily is S80°W.   What is the distance from Emily to Rachel?

Solution:

Probably the most difficult part is to drawing a picture of the problem:

Problem:

Jill, a surveyor, needs to approximate the area of a piece of land.  She walks the perimeter of the land and measures the side distances and one angle, as shown below.  What is the area of the piece of land?

Solution:

Understand these problems, and practice, practice, practice!

Use the Right Triangle Button on the MathType keyboard to enter a problem, and then click on Submit (the arrow to the right of the problem) to solve the problem. You can also click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Polar Coordinates, Equations and Graphs  – you’re ready!