Decoding the Ambiguous Case: Mastering the Law of Sines

Algebra 2 Grades High School 8:23 Video
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Lesson Description

Explore the ambiguous case (SSA) of the Law of Sines, learn to determine the number of possible triangles, and solve for missing sides and angles when two triangles exist.

Video Resource

Law of Sines (SSA) Ambiguous Case 2 Triangles Possible

Mario's Math Tutoring

Duration: 8:23
Watch on YouTube

Key Concepts

  • Law of Sines
  • Ambiguous Case (SSA)
  • Altitude of a Triangle
  • Obtuse vs. Acute Angles
  • Isosceles Triangles

Learning Objectives

  • Identify the conditions that lead to zero, one, or two possible triangles in the SSA case.
  • Calculate the altitude of a triangle to analyze the ambiguous case.
  • Apply the Law of Sines to solve for missing angles and sides in triangles, including cases with two possible solutions.
  • Solve triangles in the ambiguous case, finding all possible solutions for sides and angles.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the Law of Sines. Briefly discuss how it's used to solve triangles when given different information (e.g., ASA, AAS, SSA). Introduce the concept of the ambiguous case (SSA) and explain why it's called 'ambiguous'.
  • Video Viewing & Note-Taking (15 mins)
    Play the video 'Law of Sines (SSA) Ambiguous Case 2 Triangles Possible' by Mario's Math Tutoring. Instruct students to take notes on the different scenarios (0, 1, or 2 triangles) and the conditions that determine each case. Emphasize the importance of calculating the altitude.
  • Discussion: Identifying the Number of Possible Triangles (10 mins)
    Lead a class discussion focusing on the rules for determining the number of possible triangles in the SSA case. Use examples to illustrate each scenario. Ask students to explain in their own words how the side lengths and altitude relate to the number of possible triangles.
  • Worked Example: Solving the Two-Triangle Case (20 mins)
    Reiterate the example from the video where two triangles are possible. Work through the steps of using the Law of Sines to find the missing angles and sides for both triangles. Emphasize the use of the sine inverse function and how to find the obtuse angle solution.
  • Practice Problems (15 mins)
    Provide students with practice problems involving the ambiguous case. Include problems where they need to determine if there are 0, 1, or 2 possible triangles, and then solve for all missing sides and angles.

Interactive Exercises

  • Triangle Construction
    Give students different SSA scenarios and have them try to physically construct the triangles using rulers and protractors. This will help them visualize why some scenarios result in multiple solutions or no solutions.
  • Calculator Exploration
    Have students explore what happens when they try to use the Law of Sines in a case where no triangle is possible (e.g., the calculator returns an error). Discuss why this happens mathematically.

Discussion Questions

  • Why is the SSA case called the 'ambiguous case'?
  • How does the altitude of a triangle help determine the number of possible triangles in the SSA case?
  • Explain the steps involved in solving a triangle when there are two possible solutions.
  • When finding an angle using the inverse sine function, how do you determine the possible obtuse angle solution?

Skills Developed

  • Applying the Law of Sines
  • Trigonometric Problem Solving
  • Logical Reasoning
  • Critical Thinking
  • Spatial Visualization

Multiple Choice Questions

Question 1:

In the SSA case, which of the following conditions indicates that there is NO triangle possible?

Correct Answer: The side opposite the given angle is shorter than the altitude.

Question 2:

What does SSA stand for in the context of solving triangles?

Correct Answer: Side-Side-Angle

Question 3:

If the side opposite the given acute angle in an SSA triangle is longer than the adjacent side, how many triangles are possible?

Correct Answer: 1

Question 4:

What trigonometric function is primarily used in the Law of Sines?

Correct Answer: Sine

Question 5:

In an SSA case, if the side opposite the given acute angle is in between the altitude and the adjacent side, how many triangles are possible?

Correct Answer: 2

Question 6:

What is the first step when trying to solve an SSA triangle?

Correct Answer: Find the altitude

Question 7:

If you calculate the sine of an angle and the value is greater than 1, what does this indicate?

Correct Answer: There is no triangle possible

Question 8:

What is the property of isosceles triangles that is useful when solving the ambiguous case?

Correct Answer: Base angles are congruent

Question 9:

When solving for an angle using the inverse sine function, how can you find a possible obtuse angle solution?

Correct Answer: Subtract the acute angle from 180 degrees

Question 10:

If the side opposite the given acute angle is the EXACT length as the altitude, how many triangles are possible?

Correct Answer: 1 (right triangle)

Fill in the Blank Questions

Question 1:

The SSA case is also known as the ______ case.

Correct Answer: ambiguous

Question 2:

To determine the number of possible triangles in the SSA case, it is important to find the _______ of the triangle.

Correct Answer: altitude

Question 3:

If the side opposite the given angle is longer than the adjacent side (in the SSA case with an acute angle), there is only _____ triangle possible.

Correct Answer: one

Question 4:

When you try to solve a triangle with no solution, your calculator may display an ______ message.

Correct Answer: error

Question 5:

If a triangle has two sides of equal length, it is called an _________ triangle.

Correct Answer: isosceles

Question 6:

In an isosceles triangle, the angles opposite the equal sides are ________.

Correct Answer: congruent

Question 7:

The Law of _______ is used to solve for missing sides and angles in non-right triangles.

Correct Answer: Sines

Question 8:

If the side opposite a given acute angle is between the altitude and the adjacent side, there are ________ possible triangles.

Correct Answer: two

Question 9:

When using the inverse sine function to find an angle, remember to consider the ________ angle as a possible solution.

Correct Answer: obtuse

Question 10:

A line segment from a vertex perpendicular to the opposite side is called the _______.

Correct Answer: altitude

Educational Standards

A2.F.1:Understand functions as descriptions of covariation (how related quantities vary together).

Teaching Materials

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