Oblique Triangles

Mr. Fowler's Homepage Assignments Trig Basics Graphing Applications Oblique Triangles Sum/Diff

Oblique Triangles

Jump to: Law of Sines Ambiguous Case Law of Cosines Area

We retain our theme that Algebra is Transformations: Turning difficult-looking problems into ones we have an idea of how to solve. Since, so far, we really only know how to solve Right Triangles, it would sure be nice if we could convert Oblique (non-right) Triangles into them. In fact, we can:

create right triangles by dropping an altitude from one of the angles, then
use the Pythagorean Theorem and SOH-CAH-TOA (the definitions of the 3 basic trig functions) to find ways or formula to help along the way.

Before starting, though, we need to remember a little geometry:

Working with triangles, we used some 3-letter shortcuts to describe what we needed to know in order to prove congruence. We will use those same 3-letter shortcuts to describe what we need to know about our oblique triangles in order to determine which solution process to use.
The order of those letters was and still is important; e.g., ASA indicates the side INCLUDED between 2 angles.

Law of Sines - AAS and ASA

The Law of Sines works when we know 2 angles and a side opposite one of those angles. Of course, in a triangle, if we know 2 angle measures, we really know all three (Angle Sum Theorem).
The Law of Sines worksheet walks us through a derivation of the Law of Sines formula.

First, we drop an Altitude to one of the unknown sides, creating 2 smaller right triangles in our diagram.
Notice that our Known Side is the hypotenuse of one of those right triangles, while the Altitude is opposite a known angle.

So, we use the Sine function to find the value of the Altitude.

Looking now at the other right triangle, an Unknown Side is its hypotenuse, while the Altitude is opposite of the other known angle.

The Sine function again allows us to find the side we want.

Notice that Altitude is in both of the Sine equations we used above. By substituting one equation into the other, we find the general form of the Law of Sines formula.

The Practice Worksheet allows us to set the known values, calculate the others, and check our answers.

Ambiguous Case - SSA

In Geometry, SSA didn't work to prove triangles congruent. Remember why?
For us, SSA is a special case for the Law of Sines - special for the same reasons it doesn't work for congruence.
Notice that SSA provides us with a pair of opposites (side and angle) on which to use the Law of Sines.

The Ambiguous Case worksheet allows us to experiment with angles and side lengths to discover how many triangles should be available that meet our input criteria.
The second Ambiguous Case worksheet lets us see more clearly how to determine the number of triangles we should expect to solve for.
The Practice worksheet allows us to set side and angle measures, determine the number of triangles, solve for them, then check our answers.

Law of Cosines - SAS and SSS

The Law of Cosines relies heavily on the Pythagorean Theorem.
As above, we try to turn our problem into a Right Triangle Trig problem by dropping and Altitude to one of our Known Sides. However, using the Sine function as we did above does not get us to a solution (no opposites!). Instead:
- We use the Altitude to cut the Known Side into 2 segments of unknown length: (x) and (side-x),
- We can now set up a Pythagorean equation for each of the 2 small right triangles,
- After multiplying out (c-x)(c-x), we find a set of terms common to both equations: (x^2 + Alt^2),
- Substituting, we combine the 2 equations, then
- Use the Cosine function to solve for one of the side parts that we cut.
- It might read confusingly, but the result is reasonably easy to remember.

When we use the Law of Cosines, our result gives us a pair of opposites; so, we have options for finding the other parts of our triangle:

We can use the Law of Cosines a second and third time, or
We can use the Law of Sines and the Angle Sum Theorem. Many people find these calculations easier. But, we need to remember one large caution:

Use the Law of Cosines to find the largest angle first (the one opposite the longest side). To understand why, we should work with an obtuse triangle, find the smallest angle with LOC, then the largest with LOS and compare the results with using LOC on the largest angle. Consider what we already know about the LOS to figure out what happened.

The Law of Cosines worksheet takes us through a couple of long steps to see the development of the formula.
The SAS Practice worksheet allows us to set 2 sides and their included angle, then solve and check our answers.
The SSS Practice worksheet allows us to set all 3 sides, then solve for the angles and check our answers.

Area

From Geometry, we all learned that for triangles, Area = ½bh. With right triangles, it's easy since the two legs are the base and the height. But, what about oblique triangles? From the work we've done above, we can know and choose from any of the three sides for the base. We then need only find the height (Altitude) from the base.
The simplest case is knowing SAS:

Pick one known side as the Base.
Drop an Altitude to it from the angle opposite. The Altitude will be opposite the known angle.
The other known side is the hypotenuse of the small right triangle we've created.
The Law of Sines allows us to quickly calculate the Altitude = Known Side * sin(Angle).
Solving for Area, we get the equation:

Area = ½ * Side1 * Side2 * sin(Angle)

We need to be careful. This equation looks very similar to part of the Law of Cosines equation and we can be easily confused or use the wrong one.

From other starting information, we can use either the Law of Sines or the Law of Cosines to find a needed side or angle to fit into the equation above. There are other formulas we could memorize and use, too; but, that's up to each person.

The Area construction helps us experiment with calculating the Altitude and the Area of triangles.

Remember, practice makes permanent. Practice and learn now, so the quizzes and tests are easy.

Mr. Fowler's Homepage Assignments Trig Basics Graphing Applications Oblique Triangles Sum/Diff