# Law of Sines and Cosines, and Areas of Triangles

## 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; it’s unitless, since it’s basically a ratio.

We use the Law of Sines and Law of Cosines to “solve” triangles (find missing angles and sides) for oblique triangles (triangles that don’t have a right angle). This can a little complicated, since we have to know which angles and sides we do have to know which of the “laws” to use. (Note that the Law of Sines can still be used to solve right triangles, using the 90° angle, which has a sin of 1, as one of the angles.)

## Law of Sines

Again, we use the Law of Sines (or Sine Rule) to set up proportions to get other parts of a triangle that isn’t necessarily a right triangle.

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) (remember that these are “in a row” or adjacent parts of the triangle). 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; in these cases, use the Law of Cosines.

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, an acute (each angles measures less than 90°), or an obtuse (one of the angles measures more than 90°). For these cases, we have to account for both those shapes, so upon further study, we may have one triangle, two triangles with two sets of answers for each triangle, or no triangle. 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. Here is a summary for both Law of Sines and Law of Cosines:

Remember that if we have two of the three angles, we can obtain the third angle from Geometry (the sum of angles in a triangle is 180°).

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”.

When 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.

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, instead of retyping 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.

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

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 (even if that angle is B or C), so I can see what’s going on. If the side directly across from this angle (the paired side) is less than the side touching this angle, we will probably have an ambiguous case (or may have no triangle that can be formed), since the triangle could either be acute or obtuse.

In this situation, if we get an error message in the calculator when trying to get the other angle using Law of Sines, or, in the case of an obtuse triangle, we get a sum of more than 180° for the triangle’s angles, there is no triangle that can be formed (and that’s the answer).

For acute triangles, 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 then solve for two different triangles (the given two sides and one angle for the two triangles will be the same). If we get 90° for the second angle, we have one right triangle. This happens when the height of the triangle equals the paired side (the side across from the known angle). 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:

## Law of Cosines

The Law of Cosines (or Cosine Rule) is a little bit more complicated, since it’s not a simple proportion.

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). (Remember that these are “in a row” or adjacent parts of the triangle). It also will work for the Side, Side, Angle (SSA), but the Law of Sines is usually taught with this case, because of the Ambiguous Case.

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. Here again is a summary for both Law of Sines and Law of Cosines:

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. 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 another example of ambiguous cases with triangles.

Here’s a tricky one where we need to use the Law of Cosines twice:

### 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): $\displaystyle \text{Area}=\frac{1}{2}bh$ or $\displaystyle \text{Area}=\frac{{bh}}{2}$.

Now that we know trig, we get the area of a triangle without having to know the altitude if we know: 1) two sides, and the angle inside the two sides (the Side-Angle-Side or SAS case), or 2) three sides of the triangle (Side-Side-Side, or SSS case). If we don’t initially have these cases, we can solve the triangle using the Law of Sines or Law of Cosines to get the angles and sides needed.

Here are the two formulas; note that one uses the sin function, and the other (Heron’s formula) uses the sides only:

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.

Here’s an example of 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”!

Here are a few problems that use the Law of Cosines for a parallelogram. 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.

Draw the parallelograms to see how to solve the problem. Remember that adjacent angles in a parallelogram add up to 180°. These are called Same Side Interior angles.

This one’s a little tricky, since we have more than one triangle. (It’s actually a trapezoid, with two parallel sides and two non-parallel sides.) 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.

### Bearing Problem

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, like when we read a map, 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 or SW, for example, the angle would be 45° south of west, or 225° clockwise from north, and so on.

Remember that each time an object changes course, you have to draw another line to the north to map its new bearing. Here are some bearing examples:

Here is a problem; we also have to remember that $\text{Distance}=\text{Rate}\,\times \,\text{Time}$, since we are given rates and times and need to calculate distances. Also, remember from Geometry that Alternate Interior Angles are congruent when a transversal cuts parallel lines. (Note that this problem is actually solved more easily using vectors here in the Introduction to Vectors section.)

Here’s one more bearing problem; note that there are probably many other ways of doing this problem, but this one works:

Here’s an Area Word Problem:

Understand these problems, and practice, practice, practice!

Use the Right Triangle Button on the Mathway 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!

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