Unlocking the Cosine Rule: Mastering Non-Right Triangle Calculations

Algebra 2 Grades High School 15:50 Video
Print Lesson Plan Watch Video

Lesson Description

This lesson delves into the Cosine Rule, a vital tool for solving non-right triangles. Students will learn to apply the formula, identify opposite angles and sides, and solve for unknown sides and angles through examples and practice problems.

Video Resource

Cos Rule Algebra 2

Kevinmathscience

Duration: 15:50
Watch on YouTube

Key Concepts

  • Cosine Rule formula: c² = a² + b² - 2ab cos(C)
  • Identifying opposite sides and angles in a triangle
  • Applying the Cosine Rule to find unknown sides
  • Applying the Cosine Rule to find unknown angles

Learning Objectives

  • Students will be able to state and explain the Cosine Rule formula.
  • Students will be able to identify opposite sides and angles in any triangle.
  • Students will be able to apply the Cosine Rule to calculate the length of an unknown side given two sides and the included angle.
  • Students will be able to apply the Cosine Rule to calculate the measure of an unknown angle given three sides.

Educator Instructions

  • Introduction (5 mins)
    Briefly review right-triangle trigonometry (SOH CAH TOA) and highlight its limitations. Introduce the need for the Cosine Rule when dealing with non-right triangles. Show the video.
  • Cosine Rule Formula and Identification (10 mins)
    Write the Cosine Rule formula on the board: c² = a² + b² - 2ab cos(C). Explain each variable. Emphasize that 'C' is the angle opposite side 'c'. Work through examples, showing how to identify the angle and opposite side. Use example triangles with different orientations.
  • Solving for Unknown Sides (15 mins)
    Work through example problems from the video where students are asked to find the length of a side, given two sides and an included angle. Stress the importance of correct substitution into the formula. Include examples with different side/angle labeling to reinforce understanding. Use the calculator to solve and review the square root step.
  • Solving for Unknown Angles (15 mins)
    Work through example problems where students are asked to find the measure of an angle, given three sides. Demonstrate the algebraic manipulation required to isolate cos(C). Emphasize the use of the inverse cosine function (arccos or cos⁻¹) on the calculator. Reiterate the concept of order of operations when simplifying. Address the common mistake of combining terms incorrectly.
  • Practice Problems and Wrap-up (10 mins)
    Assign practice problems for students to work on individually or in pairs. Circulate to provide assistance. Review the key concepts and address any remaining questions. Emphasize the importance of careful problem-solving and checking answers.

Interactive Exercises

  • Triangle Solver
    Provide students with incomplete triangle information (e.g., two sides and an angle) and have them use the Cosine Rule to find the missing side or angle. Vary the given information to challenge their understanding.

Discussion Questions

  • When would you use the Cosine Rule instead of basic trigonometric ratios (SOH CAH TOA)?
  • Explain why it is important to correctly identify the angle opposite a side in the Cosine Rule.
  • What are some real-world applications where the Cosine Rule might be useful?

Skills Developed

  • Algebraic manipulation
  • Problem-solving
  • Trigonometric application
  • Calculator proficiency

Multiple Choice Questions

Question 1:

The Cosine Rule is used to solve what type of triangles?

Correct Answer: Non-right triangles

Question 2:

In the Cosine Rule formula, c² = a² + b² - 2ab cos(C), what does 'C' represent?

Correct Answer: The side opposite angle C

Question 3:

Given sides a = 5, b = 7, and angle C = 60 degrees, what is the approximate length of side c?

Correct Answer: 6.25

Question 4:

To find an unknown angle using the Cosine Rule, which trigonometric function do you use after isolating the cosine term?

Correct Answer: Inverse Cosine

Question 5:

If you are given three sides of a triangle, can you always use the Cosine Rule to find any of the angles?

Correct Answer: Yes

Question 6:

What is the first step when trying to find a missing side?

Correct Answer: State the Law of Cosines

Question 7:

If a = 8, b = 5, and c = 7, then what is the measure of angle C?

Correct Answer: 60 degrees

Question 8:

What is a possible real world use of the law of cosines?

Correct Answer: finding distance on a map

Question 9:

The Law of Cosines is especially useful in situations where you know:

Correct Answer: two sides and the included angle

Question 10:

Which of the following is not a component of the Law of Cosines formula?

Correct Answer: square root

Fill in the Blank Questions

Question 1:

The Cosine Rule states: c² = a² + b² - 2ab cos(_ _ _)

Correct Answer: C

Question 2:

In the Cosine Rule, 'c' represents the side ______ the angle C.

Correct Answer: opposite

Question 3:

If you are solving for an angle using the Cosine Rule, you need to use the ______ cosine function.

Correct Answer: inverse

Question 4:

When using the Cosine Rule to find a side, after calculating c², you must take the _______ to find c.

Correct Answer: square root

Question 5:

Before using the inverse cosine to solve for an angle, you must ______ cos(C) in the equation.

Correct Answer: isolate

Question 6:

The Law of Cosines is primarily used for ______ triangles.

Correct Answer: non-right

Question 7:

In the formula c² = a² + b² - 2ab cos(C), the sides a and b are always ______ to angle C.

Correct Answer: adjacent

Question 8:

To find a missing side using the Law of Cosines, you need to know the lengths of ______ other sides and ______ angle between them.

Correct Answer: two

Question 9:

The term 'included angle' refers to the angle ______ between two known sides.

Correct Answer: located

Question 10:

If three sides of a triangle are known, the Law of Cosines can be used to determine the measure of any of the ______ of the triangle.

Correct Answer: angles

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.A.2:Generate and evaluate equivalent algebraic expressions and equations using various strategies.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz Export to Google Docs
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans