Grade 10 Math | Solving Oblique Triangle Using Laws of Cosines | First Term (term 1) Week 2

Introduction to the Laws of Cosines

Overview of the Lesson

• This video tutorial focuses on Grade 10 Mathematics, specifically discussing the laws of cosines for solving oblique triangles. The content is part of the revised K10 curriculum.

• The instructor encourages viewers to subscribe for more lessons covering all topics in Grade 10 Mathematics.

Learning Objectives

• By the end of this lesson, students will be able to:

• Recall what an oblique triangle is.

• Define the law of cosines.

• Determine given measures in an oblique triangle.

• Apply laws of cosines in solving oblique triangles.

Understanding Oblique Triangles

Definition and Types

• An oblique triangle is defined as any triangle that does not contain a right angle (90°). Examples include obtuse and acute triangles.

• The instructor previously discussed oblique triangles in a prior video about the law of sines, providing foundational knowledge for this lesson.

Law of Cosines Explained

Key Concepts

• The law of cosines states that the square of a side in a triangle equals the sum of the squares of the other two sides minus twice their product times the cosine of their included angle. This can be expressed with formulas involving angles A, B, and C corresponding to sides a, b, and c respectively.

Application Cases

• There are two specific cases where the law can be applied:

• SAS (Side-Angle-Side): Two sides and their included angle are known.

• SSS (Side-Side-Side): All three sides are known. If neither case applies, then cosine law cannot be used for solving an oblique triangle.

Formulas for Law of Cosines

Formula Breakdown

• Three variations exist based on which side is being calculated:

• For side a: a^2 = b^2 + c^2 - 2bc cdot cos(A)

• For side b: b^2 = a^2 + c^2 - 2ac cdot cos(B)

• For side c: c^2 = a^2 + b^2 - 2ab cdot cos(C)

• Each formula follows a consistent pattern linking opposite angles to their respective sides. Understanding this pattern aids memorization.

Solving Using SAS Case

Example Problem Setup

• In an example using SAS case:

• Given: Side AB = 20 cm, Side AC = 15 cm, Angle B = included angle at vertex B.

• Students learn how to identify when to apply cosine law based on provided measurements. If only two sides or one angle were given without satisfying SAS or SSS conditions, cosine law would not apply here.

Calculating Missing Parts

Step-by-Step Calculation

1. Identify unknown parts: Angles A & C and Side B need calculation.

2. Start by calculating Side B using its formula derived from SAS case setup.

3. Substitute known values into formula:

[

B^2 = A^2 + C^2 − (constant)(textcosineB)

]

4. Simplify calculations stepwise until reaching numerical results for each variable involved including necessary trigonometric functions like cosine at specified angles such as Angle B being equal to 110^circ.

Finding Angles A & C

Continuing Calculations

1. After finding Side B's measurement (12.407), proceed with calculating Angle A using its respective formula linked through cosine relationships established earlier.

Final Steps

1. Use inverse trigonometric functions on your calculator after isolating terms related directly to Angle A's equation until arriving at its value approximately equal to 27.70^circ.

Calculate Angle C

1. With both Angles A & B determined already (110^circ), use properties that state all angles within any triangle must sum up to 180^circ. Thus calculate Angle C by subtracting from total degrees available within triangular geometry leading you towards final conclusions regarding missing parts within your original problem statement concerning Triangle ABC configuration outlined initially throughout this session!

Conclusion and Exercises

Practice Problems

• Viewers are encouraged to solve additional exercises presented at video's end focusing on identifying unknown parts within various triangles utilizing learned principles effectively while reinforcing understanding gained during instructional segments throughout this tutorial session! Suggestions include Triangle RST needing Side S solved alongside Triangle FPJ requiring calculation around Angle P specifically noted down below! Good luck!