Conquering Non-Right Triangles: Mastering the Law of Cosines

PreAlgebra Grades High School 5:44 Video
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Lesson Description

Learn how to apply the Law of Cosines to solve for missing sides and angles in non-right triangles. This lesson covers when to use the Law of Cosines, its various formulas, and provides practical examples.

Video Resource

Law of Cosines

Mario's Math Tutoring

Duration: 5:44
Watch on YouTube

Key Concepts

  • Law of Cosines formula
  • Side-Angle-Side (SAS) case
  • Side-Side-Side (SSS) case
  • Solving for missing sides
  • Solving for missing angles

Learning Objectives

  • Students will be able to identify when to use the Law of Cosines versus the Law of Sines.
  • Students will be able to apply the Law of Cosines to solve for missing sides in triangles when given Side-Angle-Side (SAS) information.
  • Students will be able to apply the Law of Cosines to solve for missing angles in triangles when given Side-Side-Side (SSS) information.
  • Students will understand how the arrangement of sides and angles relates to the appropriate formula

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the limitations of using standard trigonometric ratios (SOH CAH TOA) for non-right triangles. Introduce the Law of Cosines as a tool to overcome these limitations. Briefly mention the Law of Sines and differentiate when each law should be used (Law of Cosines when you have SAS or SSS, Law of Sines when you have ASA or AAS).
  • Understanding the Law of Cosines Formulas (10 mins)
    Present the three forms of the Law of Cosines formulas, emphasizing the pattern and relationship between sides and angles. Explain how each formula is derived from the others by simply rearranging the letters. c² = a² + b² - 2ab cos(C), a² = b² + c² - 2bc cos(A), b² = a² + c² - 2ac cos(B). Highlight that the angle used in the cosine function is always opposite the side isolated on the other side of the equation.
  • Example 1: Solving for a Missing Side (SAS) (15 mins)
    Work through the first example from the video (1:25 timestamp). Clearly demonstrate each step of the process, including substituting known values into the correct formula, simplifying the equation, and solving for the unknown side. Emphasize the order of operations. Remind students that the cosine of an angle is just a number, not a variable. Show students the importance of being careful with order of operations.
  • Example 2: Solving for a Missing Angle (SSS) (15 mins)
    Work through the second example from the video (3:17 timestamp). Highlight the algebraic manipulation required to isolate the cosine term and then use the inverse cosine function to find the angle. Stress the importance of not combining terms incorrectly (e.g., subtracting from within the cosine term). Review the concept of inverse trigonometric functions.
  • Wrap-up and Application (5 mins)
    Summarize the key concepts and provide additional real-world examples where the Law of Cosines might be applied (e.g., navigation, surveying). Mention using the Law of Sines after finding one missing angle when given SSS. Briefly introduce the concept of ambiguous case when using the Law of Sines

Interactive Exercises

  • Practice Problems
    Provide students with a set of practice problems involving both SAS and SSS scenarios. Encourage them to work independently or in small groups. Offer guidance and feedback as needed.
  • Real-World Applications
    Present a real-world scenario (e.g., finding the distance between two ships given their bearings and distances from a common point) and have students apply the Law of Cosines to solve it.

Discussion Questions

  • When would you use the Law of Cosines instead of the Law of Sines?
  • How does the Law of Cosines relate to the Pythagorean theorem?
  • Can you use the Law of Cosines to solve a right triangle? If so, how?

Skills Developed

  • Trigonometric problem-solving
  • Algebraic manipulation
  • Application of mathematical formulas
  • Critical thinking

Multiple Choice Questions

Question 1:

The Law of Cosines is most directly applicable when you are given which of the following?

Correct Answer: Side-Angle-Side (SAS)

Question 2:

Which formula correctly represents the Law of Cosines when solving for side 'a'?

Correct Answer: a² = b² + c² - 2bc cos(A)

Question 3:

In a triangle with sides a=5, b=7, and c=8, which formula would you use to find angle A?

Correct Answer: cos(A) = (7² + 8² - 5²) / (2 * 7 * 8)

Question 4:

If you are given three sides of a triangle and use the Law of Cosines to find one angle, what law can you then use to find the remaining angles?

Correct Answer: Law of Sines

Question 5:

What is the first step in solving for an angle using the Law of Cosines when given SSS?

Correct Answer: Calculate the sine of the angle.

Question 6:

Given a triangle with sides a=3, b=4 and angle C = 60 degrees, what is the value of c²?

Correct Answer: 13

Question 7:

Using the information from the previous question, what is the value of side 'c'?

Correct Answer: √13

Question 8:

When solving for an angle with the Law of Cosines, the last step involves applying what?

Correct Answer: Inverse Cosine Function

Question 9:

Which of the following is a limitation of only knowing Side-Side-Side?

Correct Answer: Cannot determine the angles initially

Question 10:

The Law of Cosines is a generalization of which theorem?

Correct Answer: Pythagorean Theorem

Fill in the Blank Questions

Question 1:

The Law of Cosines is used to solve ________ triangles.

Correct Answer: non-right

Question 2:

When using the Law of Cosines to find a missing angle, you must use the ________ cosine function as the final step.

Correct Answer: inverse

Question 3:

The acronym SAS stands for ________, Angle, ________.

Correct Answer: Side

Question 4:

If you know all three sides of a triangle, you can find any ________ using the Law of Cosines.

Correct Answer: angle

Question 5:

If you know two sides and the included angle of a triangle, you can find the missing ________ using the Law of Cosines.

Correct Answer: side

Question 6:

c² = a² + b² - 2ab cos(C) is the formula for ________.

Correct Answer: Law of Cosines

Question 7:

After finding one angle in a triangle using the Law of Cosines, you can use the Law of ________ to solve for other angles.

Correct Answer: Sines

Question 8:

Given SSS, if the result from using the Law of Cosines on one angle produces an 'undefined' result, then that value represents ________.

Correct Answer: an invalid triangle

Question 9:

Before applying the Law of Cosines formula, it's crucial to correctly identify the sides and ________.

Correct Answer: angles

Question 10:

The law of cosines is most helpful when using ________ and ________, where the letters represent sides and angles.

Correct Answer: SSS, SAS

Educational Standards

PC.T.2.2:Apply the Law of Sines and Law of Cosines to solve problems. PC.T.2.1:[Objective] Create models for situations involving trigonometry.

Teaching Materials

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