The Law of Cosines Study Guide

Standards for Mathematical Content and Practice

Standards for Mathematical Content

• G.SRT.10: Prove the Laws of Sines and Cosines and use them to solve problems.

• G.SRT.11: Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.

Standards for Mathematical Practice

• MP3: Construct viable arguments and critique the reasoning of others.

• MP6: Attend to precision.

Lesson Goal

The primary objective is to understand and apply the Law of Cosines to determine unknown measurements in both right and non-right (oblique) triangles.

Fundamentals of the Law of Cosines

Overview

The Law of Cosines is a critical trigonometric tool used when the Law of Sines is insufficient for solving a triangle. It provides a method to calculate:

1. The length of the third side of a triangle when the measures of two sides and the included angle (SAS - Side-Angle-Side) are known.

2. The measures of the angles of a triangle when the lengths of all three sides (SSS - Side-Side-Side) are known.

Theorem 9.11: The Law of Cosines

In any triangle $\triangle ABC$, let $a$, $b$, and $c$ represent the lengths of the sides opposite angles $A$, $B$, and $C$, respectively. The Law of Cosines states the following relationships:

• $a^{2} = b^{2} + c^{2} - 2bc \cos(A)$

• $b^{2} = a^{2} + c^{2} - 2ac \cos(B)$

• $c^{2} = a^{2} + b^{2} - 2ab \cos(C)$

Application to Right Triangles

Talk About It!: A specific inquiry is posed regarding the result of applying the Law of Cosines to a right triangle. If the included angle is $90^{\circ}$, the term containing the cosine becomes zero because $\cos(90^{\circ}) = 0$. Consequently, the formula reduces to the Pythagorean Theorem: $c^{2} = a^{2} + b^{2}$.

Detailed Problem-Solving Procedures

Example 1: The Law of Cosines (SAS)

When given the measures of two sides and their included angle, use the following procedure to find the missing side length:

1. Identify Given Info: Side lengths and the measure of the included angle.

2. Write Equation: Select the version of the Law of Cosines corresponding to the unknown side.

3. Substitution: Insert the known side lengths and the angle measure into the formula.

4. Simplify: Calculate the squares of the sides and the product of the numerical terms.

5. Isolate Variable: Take the positive square root of each side of the equation.

6. Final Computation: Use a calculator to determine the final approximate length.

Example 2: The Law of Cosines (SSS)

When the lengths of all three sides are known, the Law of Cosines is used to find missing angle measures:

1. Write Equation: Use the Law of Cosines formula that includes the angle you wish to find.

2. Substitution: Substitute the three known side lengths into the equation.

3. Simplify: Perform the arithmetic for the squared terms and the product term.

4. Solve for Cosine: Isolate the cosine term.

5. Inverse Function: Use the inverse cosine function ($\cos^{-1}$) to solve for the angle value.

6. Final Computation: Use a calculator to find the degree measure.

Example 3: Indirect Measurement

This practical application involves finding distances across obstacles where direct measurement is impossible.

• Scenario: Use the measures of two sides and the included angle (SAS) within a mapped triangle.

• Procedure:   - Apply the Law of Cosines to set up the equation for the unknown distance.   - Simplify the squared values and products.   - Take the positive square root.   - Calculate the final value using a calculator to determine the ground distance.

Example 4: Solving a Non-right Triangle

To fully "solve" a triangle means to find all missing sides and angles. This often requires combining the Law of Cosines and the Law of Sines:

1. Find the first unknown angle: If the triangle is SSS, use the Law of Cosines first. (Note: Specific advice suggests finding the largest angle first using the Law of Cosines).

2. Find the second unknown angle: Use the Law of Sines to find a second angle measurement.    - Step: Apply the Multiplication Property of Equality to isolate the sine of the angle.    - Step: Use the inverse sine function ($\sin^{-1}$) and a calculator.

3. Find the third angle: Subtract the sum of the first two angles from $180^{\circ}$.

Example 5: Solving a Right Triangle with Law of Cosines

While specialized right-triangle trigonometry (SOH CAH TOA) or the Pythagorean Theorem are standard, the Law of Cosines remains valid for right triangles:

• Side Calculation: Use the Pythagorean Theorem ($a^{2} + b^{2} = c^{2}$), substitute values, simplify, and take the square root.

• Angle Calculation: Law of Cosines can be applied by substituting the sides; however, standard right-triangle trigonometry is usually more efficient.

Summary: Triangle Solvability (Exit Ticket)