# Law of Sines Ambiguous Case Worksheets

I find this topic to have problems that seem like riddles or puzzles, by the have a great deal of application in the real world. There will be situations that arise like this on construction sites everywhere. In this topic we are applying the use of SSA. In certain circumstances it can require multiple solutions or no solution at all. We begin by working off of the pretense of breaking a large triangle into multiples. These worksheets and lessons help students learn how to manipulate the use of the law of sines to determine missing measures of triangles when you run into an ambiguous case.

### Aligned Standard: HSG-SRT.D.10

- Triangle Challenge Step-by-Step Lesson- You're given some missing data about a triangle and then ask to determine the number of triangles you can break into that area.
- Guided Lesson - When I first presented this years ago people thought I was nuts, but after they do it once; they like it.
- Guided Lesson Explanation - These problems are like card tricks. You feel like your in "the know", when you understand them.
- Practice Worksheet - I gave you a diagram on a couple of the problems to help you set them up.
- Matching Worksheet - Match those values to the missing parts of the triangles and don't get throw by a and e.
- Find the Sine of One Point Worksheet Five Pack - These are much easier if you crossed out two pieces of information. Any idea which?
- Sine: Find the Value of x Worksheet Five Pack - Looking for the measures of sides and angles.
- Using a Calculator (Inverse Function) Worksheet Five Pack - Time to learn how to use the inverse function part of your calculator.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

These problems are really neat. You need to find out how many triangles you can make from the givens.

- Homework 1 - If we use the reference angle 44° in Quadrant II, the angle C is 136°.
- Homework 2 - Use the Law of Sines: a/sin A = c/sin c
- Homework 3 - With m ∠ A = 60° and m ∠ C =. 137° the sum of the angles would exceed 180°.

### Practice Worksheets

A rough diagram is provided to help students focus on the concept skills.

- Practice 1 - How many distinct triangles can be drawn given these measurements?
- Practice 2 - m ∠ A = 58° a = 12 c = 6 .
- Practice 3 - Since sin C must be < 1, no angle exists for angle C. No triangles exists for these measurements.

### Math Skill Quizzes

Find all the missing pieces and parts using geometry.

- Quiz 1 - From the diagram solve the following: m < A = 65°, a = 17 and b = 16
- Quiz 2 - Calculate the value of sin
^{-1}0.29 - Quiz 3 - Find tan X. When you are given the value of the sides: YX = 8, XZ = 4, and YZ = 2.

### What is the Ambiguous Case of the Law of Sines?

When you are using the Law of Sines to find a missing angles within a triangle, you will run into situations where you could create two completely different triangles based on the information that is being presented to you. Up until now, every time we have used one of these theorems to determine missing measures there has always been a single solution. Normally in these situations we would use the SSA theorem to find that value, but since this calls for alternate interpretations of what is available, this does not apply. This leads to one of several different scenarios. Hence, why this is called “Ambiguous”. Ambiguous means open to interpretation. The triangle based on the given information does not exist. There can also be a situation where two separate triangles could possibly be formed. In this case we will determine the solution twice, one for each missing of the two possible triangles.

There are three different scenarios that can result when you come across this. We can prove that no triangle exists and that does not require a solution. You can have a single triangle present those results in a single solution. There is also the possibility of two triangles being present and as a result there are two possible solutions. There are several facts that we know about triangles that helps us determine which of these applies. The two that I find we most commonly use to determine this is that the sum of the interior angles are 180 degrees, and no triangle can have two obtuse angles. It also helps to know that a right angle in a triangle has to be its largest angle.

### Tips for Solving Trigonometry Problems

Students often get to solving the Trigonometry Problems stage when they are in the ninth grade and it can get tricky at times. In this topic, we will be covering a general or basic idea regarding Solving Trigonometry Problems along with some useful tips. In mathematics, it is essential to understand how you understand something rather than memorizing the steps. Trigonometry is the study of triangles. Let's discuss some of the tips. 1. The first step involves remembering the formulas and definitions. Unless and until you are familiar with the identities and the background information of a trigonometric problem, till then, you cannot get better at Solving Trigonometry Problems. 2. The second tip is practice. The real reason why most students struggle with solving trigonometric problems is because of a lack of practice. Learning the formulas is the easier part; the bigger challenge is to maintain the continuous practice of every single formula and learning variations of problems. 3. Practice your way into difficulty. If you are getting too comfortable with a particular level of difficulty, then it is recommended you increase the level and do more difficult ones.