Up to this point, we have only looked at how to solve Right Triangles, but in this lesson, we are going to explore the Law of Sines and how to solve Oblique Triangles.
Oblique Triangles!
What are those?
An Oblique Triangle is any triangle that is not a Right Triangle.
Ok, that’s cool, but what do you mean by “solving a triangle?”
Well, when we have a Right Triangle, we utilized the Pythagorean Theorem and SOH-CAH-TOA to help us solve for any missing sides or angles.
So, if we have an Oblique Triangle, and we too want to solve for any missing sides or angles, we will need draw upon our previous knowledge of:
- Sum of the interior angles of any triangle is 180 degrees.
- How to determine congruence (ASA, SSA, SAS, AAS, SSS).
And learn two additional tools:
- Law of Sines
- Law of Cosines
Don’t worry, while it may have been a while since you’ve see those Congruency Axioms that you discovered back in Geometry, instead of using them to prove two triangles are the same, or congruent, we are going to merely use them to help us choose the right formula!
When to Use the Law of Sines
The Law of Sines is utilized whenever you have either Angle-Side-Angle (ASA) or Angle-Angle-Side (AAS) congruency.
In fact, we will also learn one more type of congruency that the Law of Sines can be used in our next lesson titled the Ambiguous Case.
The trick to knowing when to use the Law of Sines is to draw a picture and determine what parts of the triangle are known and what parts are missing.
How to Use the Law of Sines
- First, we will learn how to draw and label a standard Oblique Triangle, and it will be the same type of drawing we will use for this entire unit.
- Determine the congruency from the given information, either ASA or AAS, and set up your ratios to find the missing side length or angle.
- Solve for all unknown parts of the triangle.
Wait!
What ratios?
Excellent question!
The Law of Sines definition consists of three ratios, where we equate the sides and their opposite angles.
Once we have established which ratio we need to solve, we simply plug into the formula or equation, cross multiply, and find the missing unknown (i.e., side or angle).
In addition to using cross multiplication we will also utilize inverse trig functions and our handy dandy calculator to help us find all the missing parts of a triangle.
As you will quickly see, the Law of Sines, or the Sine Rule, as Math is Fun calls it, is quick and easy to use and will be a valuable tool in your tool belt for solving triangles.
Law of Sines Worksheet
- Law of Sines Handout: This practice sheet includes the law of sines formula, steps for solving oblique triangles, and 2 practice problems with solutions.
How to Solve The Law of Sines – Video
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