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Algebra 2
Mastering Law of Sines and Cosines: A Complete Guide

Mastering Law of Sines and Cosines: A Complete Guide

Algebra 2 Grades High School 38:47 Video

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Lesson Description

This lesson plan provides a comprehensive guide to understanding and applying the Law of Sines and the Law of Cosines in solving non-right triangles. Students will learn to find missing sides, angles, and solve ambiguous case triangles.

Video Resource

Law of Sines and Law of Cosines (Complete Guide)

Mario's Math Tutoring

Duration: 38:47

Watch on YouTube

Key Concepts

Law of Sines
Law of Cosines
Ambiguous Case (SSA)
Triangle Inequality Theorem

Learning Objectives

Apply the Law of Sines to solve for missing sides and angles in non-right triangles.
Apply the Law of Cosines to solve for missing sides and angles in non-right triangles.
Identify and solve the ambiguous case (SSA) of triangles, determining if there are zero, one, or two possible solutions.
Use the triangle inequality theorem to verify if a triangle can be drawn.

Educator Instructions

Introduction (5 mins)
Begin by reviewing basic trigonometric ratios (sine, cosine, tangent) in right triangles. Introduce the concept of solving non-right triangles and the need for the Law of Sines and Law of Cosines. Briefly introduce the video by Mario's Math Tutoring.
Law of Sines (15 mins)
Play the video from the beginning to the section explaining the Law of Sines. Explain the formula and how to set up proportions to solve for missing sides and angles. Emphasize using pairs of angles and their opposite sides. Work through example problems from the video and provide additional examples.
Law of Cosines (15 mins)
Play the video section explaining the Law of Cosines. Explain the formula and when to use it (Side-Angle-Side or Side-Side-Side). Discuss the different forms of the Law of Cosines and the relationship between the angle and the side opposite. Work through example problems from the video and provide additional examples.
The Ambiguous Case (SSA) (20 mins)
Play the video section explaining the ambiguous case (SSA). Explain the conditions that lead to zero, one, or two possible triangles. Show how to determine the number of possible triangles by comparing the side opposite the given angle to the altitude and the adjacent side. Work through example problems from the video and provide additional examples. Emphasize that two sides must add up to more than the third side.
Complete Triangle Solving (10 mins)
Demonstrate how to completely solve a triangle by finding all missing sides and angles, using both the Law of Sines and Law of Cosines as needed. Review strategies for choosing the appropriate law to use. Work through example problems from the video and provide additional examples.
Conclusion (5 mins)
Summarize the key concepts of the Law of Sines and Law of Cosines. Review the steps for solving different types of triangle problems. Assign practice problems for homework.

Interactive Exercises

Triangle Solver
Students use an online triangle solver tool to input given information and check their work when solving triangle problems. (GeoGebra, Desmos)
Ambiguous Case Challenge
Present students with several SSA triangle problems and have them determine the number of possible solutions, justifying their answers.

Discussion Questions

When is it appropriate to use the Law of Sines versus the Law of Cosines?
How can you determine if a triangle problem has one, two, or no solutions?
What are some real-world applications of the Law of Sines and Law of Cosines?

Skills Developed

Problem-solving
Critical thinking
Algebraic manipulation
Trigonometric application

Multiple Choice Questions

Question 1:

When should you use the Law of Sines to solve a triangle?

When given three sides (SSS).

When given side-angle-side (SAS).

When you have an angle and its opposite side.

Only for right triangles.

Correct Answer: When you have an angle and its opposite side.

Question 2:

Which of the following is the Law of Cosines formula to find side 'c'?

c = a + b

c² = a² + b²

c² = a² + b² - 2ab cos(C)

c² = a² - b² + 2ab cos(C)

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

Question 3:

What is the Ambiguous Case (SSA)?

When there's only one possible triangle.

When no triangle is possible.

When given side-side-angle, there might be 0, 1, or 2 triangles.

When all angles are known.

Correct Answer: When given side-side-angle, there might be 0, 1, or 2 triangles.

Question 4:

If you are given SSS (three sides) of a triangle, what should you use?

Law of Sines

Law of Cosines

Pythagorean Theorem

SOH CAH TOA

Correct Answer: Law of Cosines

Question 5:

In triangle ABC, angle A = 30°, side a = 6, and side b = 10. How many possible triangles are there?

Cannot be determined

Correct Answer: 2

Question 6:

The triangle inequality theorem states...

All sides of a triangle must be equal.

Two sides of a triangle must be equal.

Any two sides have to add up to more than the third side

Any two sides have to add up to less than the third side

Correct Answer: Any two sides have to add up to more than the third side

Question 7:

If two angles of a triangle are 45 and 60 degrees, what is the measure of the third angle?

Correct Answer: 75

Question 8:

If the sides of a triangle are a= 5, b = 7, c = 10. What law is most appropriate to find the angles?

Law of Sines

Law of Cosines

Pythagorean Theorem

SOH CAH TOA

Correct Answer: Law of Cosines

Question 9:

How many degrees does the interior angles of a triangle add up to?

180

270

360

Correct Answer: 180

Question 10:

The longest side of a triangle is...

Across from the largest angle

Across from the smallest angle

The hypotenuse

The base

Correct Answer: Across from the largest angle

Fill in the Blank Questions

Question 1:

The Law of ______ is used when you have an angle and its opposite side.

Correct Answer: Sines

Question 2:

The Law of ______ is used when you are given SSS or SAS.

Correct Answer: Cosines

Question 3:

The SSA case is also know as the ______ case.

Correct Answer: ambiguous

Question 4:

If a + b ≤ c, where a, b, and c are the side lengths of a triangle, then ______ triangle(s) can be formed.

Correct Answer: no

Question 5:

The _______ side is always across from the largest angle.

Correct Answer: longest

Question 6:

The sine of 30 degrees is equal to _______.

Correct Answer: 0.5

Question 7:

The three angles of a triangle add up to _______ degrees.

Correct Answer: 180

Question 8:

If you have side-side-side you must start with the Law of ______.

Correct Answer: Cosines

Question 9:

If a triangle has two equal sides it is known as a _______ triangle.

Correct Answer: isosceles

Question 10:

An _______ is a line from a vertex to the opposite side, forming a right angle.

Correct Answer: altitude

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.1.1:Use mathematical models to represent quadratic relationships and solve using factoring, completing the square, the quadratic formula, and various methods (including graphing calculator or other appropriate technology). Find non-real roots when they exist. A2.F.1:Understand functions as descriptions of covariation (how related quantities vary together).

Teaching Materials

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