Mastering Triangles: Law of Sines and Cosines

Algebra 2 Grades High School 6:23 Video
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Lesson Description

Learn to solve non-right triangles using the Law of Sines and Law of Cosines. This lesson covers when to apply each law and provides step-by-step examples.

Video Resource

Solving Triangles Using Law of Sines and Law of Cosines

Mario's Math Tutoring

Duration: 6:23
Watch on YouTube

Key Concepts

  • Law of Sines
  • Law of Cosines
  • Solving Triangles (finding all angles and sides)
  • When to use Law of Sines vs. Law of Cosines
  • Ambiguous Case

Learning Objectives

  • Students will be able to apply the Law of Sines to solve for missing angles and sides in non-right triangles.
  • Students will be able to apply the Law of Cosines to solve for missing angles and sides in non-right triangles.
  • Students will be able to determine when to use the Law of Sines versus the Law of Cosines based on the given information.
  • Students will be able to solve a triangle completely, finding all angles and sides.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the properties of triangles and the differences between right and non-right triangles. Introduce the Law of Sines and the Law of Cosines as tools for solving non-right triangles. Briefly explain when each law is applicable.
  • Law of Sines Example (15 mins)
    Watch the first example from the video (solving a triangle using only the Law of Sines). Pause the video at key points to explain each step, emphasizing the cross-multiplication and solving for the unknown side. Discuss why using original values is preferred over rounded values when possible.
  • Law of Cosines Example (20 mins)
    Watch the second example from the video (solving a triangle using both the Law of Cosines and the Law of Sines). Explain why the Law of Cosines must be used first in this scenario (Side-Angle-Side). Break down the steps of applying the Law of Cosines to find the missing side. Then, show how the Law of Sines can be used to find the missing angles. Discuss using inverse sine to solve for angles and the importance of calculator mode (degrees).
  • Practice Problems (15 mins)
    Provide students with practice problems involving both the Law of Sines and the Law of Cosines. Encourage them to work in pairs or small groups. Circulate to provide assistance and answer questions.
  • Review and Closure (5 mins)
    Review the key concepts and learning objectives. Answer any remaining questions. Preview the next lesson on ambiguous cases in law of sines.

Interactive Exercises

  • Triangle Solver Worksheet
    Provide students with a worksheet containing a variety of triangle problems, some solvable by the Law of Sines, some by the Law of Cosines, and some requiring both. Students must first determine which law(s) to apply and then solve for the missing sides and angles.

Discussion Questions

  • When is it appropriate to use the Law of Sines to solve a triangle?
  • When is it appropriate to use the Law of Cosines to solve a triangle?
  • Why is it important to use original, non-rounded values when possible in these calculations?
  • What are the potential pitfalls of using rounded values in subsequent calculations?

Skills Developed

  • Problem-solving
  • Critical thinking
  • Application of formulas
  • Trigonometry
  • Algebraic manipulation

Multiple Choice Questions

Question 1:

Which law is best suited to solve a triangle when you know two angles and one side?

Correct Answer: Law of Sines

Question 2:

Which law is best suited to solve a triangle when you know all three sides?

Correct Answer: Law of Cosines

Question 3:

If you round intermediate values during calculation, what is most likely to happen?

Correct Answer: Decrease accuracy

Question 4:

What does 'solving a triangle' mean?

Correct Answer: Finding all angles and sides

Question 5:

Given angle A = 30°, side a = 5, and angle B = 45°, which formula would you use to find side b?

Correct Answer: a/sin(A) = b/sin(B)

Question 6:

Given sides a = 7, b = 8, and c = 9, which formula would you use to find angle C?

Correct Answer: c² = a² + b² - 2ab*cos(C)

Question 7:

If the mode on your calculator is set to radians but the angle is degrees, what is most likely to happen?

Correct Answer: The answer will be incorrect.

Question 8:

What information do you need to apply the Law of Sines?

Correct Answer: Two angles and any side

Question 9:

What information do you need to apply the Law of Cosines when solving for a side?

Correct Answer: Two sides and the included angle

Question 10:

Side-Side-Angle (SSA) is known as what case when dealing with the Law of Sines?

Correct Answer: Ambiguous Case

Fill in the Blank Questions

Question 1:

The Law of ______ can be used when you know two angles and a non-included side of a triangle.

Correct Answer: Sines

Question 2:

The Law of ______ is useful when you know all three sides of a triangle.

Correct Answer: Cosines

Question 3:

When solving for an angle using the Law of Sines, you may need to use the ______ sine function.

Correct Answer: inverse

Question 4:

If you know Side-Angle-Side (SAS), you must use the law of ______ first.

Correct Answer: cosines

Question 5:

The ______ case refers to the ambiguous situation when using the Law of Sines.

Correct Answer: SSA

Question 6:

The law of sines states that the ratio of a side to the ______ of its opposite angle is constant.

Correct Answer: sine

Question 7:

In the Law of Cosines formula c² = a² + b² - 2ab*cos(C), C represents the ______ opposite side c.

Correct Answer: angle

Question 8:

When solving triangles, it is important to check your calculator's ______ setting to ensure it is in degrees or radians as required.

Correct Answer: mode

Question 9:

The process of finding all angles and sides of a triangle is called ______ the triangle.

Correct Answer: solving

Question 10:

When provided two sides and the non-included angle you have to check for the ______ case.

Correct Answer: ambiguous

Educational Standards

A2.A.1:Represent and solve mathematical and real-world problems using nonlinear equations, systems of linear equations, and systems of linear inequalities; interpret the solutions in the original context. A2.F.1:Understand functions as descriptions of covariation (how related quantities vary together).

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