Study Notes on Testing, Triangle Law of Cosines, Area of Triangles and Applied Problems

Test Information

• Students have a test to take in the testing center this week.

• Reminder to check the testing center’s hours of operation.

• Testing center open on Thursday and Friday also.

• Ensure to arrange any necessary appointments, especially if visiting Eaton's, Dare, or Currituck testing centers, which may not always be open.

• Students can reach out via email for assistance if there are any questions about scheduling the test.

Class Schedule

• No class is scheduled for Thursday this week.

• Students can utilize the regular class time on Thursday to take the exam.

• The exam duration is two hours.

Material Coverage

• The session will cover all needed information and notes.

• In case some content isn’t covered today, there will be an additional class at 12:30 tomorrow.

  • The video of this class will be posted for students to access later to finish the notes.

Law of Cosines (Section 7.3)

Overview

• Continuing discussion of triangles, specifically the law of cosines.

• Recap of previous topics on solving right triangles and the law of sines.

Cases in Triangles

• Case 1 and Case 2 were discussed previously (Law of Sines).

• Today focuses on:

  • Case 3: Side-Angle-Side (SAS)

  • Given: Two sides and the included angle.

  • Case 4: Side-Side-Side (SSS)

  • Given: Three sides.

Law of Cosines Definition

• For any triangle with sides labeled a, b, and c, and respective opposite angles A, B, and C:

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

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

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

Solving Triangles Using the Law of Cosines

Example 1: Triangle with Given Sides and Angle

• Given: a = 2, b = 3, angle C = 60 degrees.

• Setup: Calculate the third side using the formula:

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

• Calculation steps:

  1. $c^2=2^2+3^2-2\cdot2\cdot3\cdot{cos}(60)$

  2. Substitute known values:

  • $c^2=4+9-2\cdot2\cdot3\cdot0.5$

  • $c^2 = 13 - 6 = 7$

  1. Thus, $c=\sqrt7$

Finding Missing Angles

• To find angle A:

  • Using the law of cosines:

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

  • Substitute:

  • $2^2=3^2+\sqrt7^2-2\cdot3\cdot\sqrt7\cdot{cos}(A)$

  • Calculate and simplify to find A.

Example 2: Triangle with All Three Sides

• Given: a = 4, b = 3, c = 6 (Case 4).

• Start by finding angle A, using:

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

  • Substitute and simplify to find angle A:

  • $16=9+36-2\cdot3\cdot6\cdot{cos}(A)$

  • Continue to isolate and compute A.

Applied Problem

• Scenario: A sailboat moves from Naples, Florida to Key West.

  • Distance = 150 miles at a speed of 15 miles/hour.

  • After 4 hours, the angle is off by 20 degrees.

Calculating Distances and Angles

1. Calculate distance traveled:

   • ${distance}={speed}\cdot{time}=15\cdot4=60_{}{ miles}.$

2. Use law of cosines to find remaining distance from Key West, drawing a triangle with appropriate side lengths and angles.

3. Distances, remaining angles, and overall calculations yield solutions for navigation and course correction.

Triangle Area (Section 7.4)

Area Formula for Triangles

• General formula: $A = rac{1}{2} imes ext{base} imes ext{height}.$

• In the case of SAS triangles,

  • When given sides a, b and included angle c:

  • Use $A = rac{1}{2}ab imes ext{sin}(c)$

Heron’s Formula for SSS Triangles

• For triangles with sides a, b, c:

  • Calculate semi-perimeter, $s = rac{1}{2}(a+b+c)$

  • Area given by, $A = ext{sqrt}(s(s-a)(s-b)(s-c)).$

Example: Area Calculation

1. Find area with sides 4, 5, and 7 using Heron's formula.

2. Calculate total area needed when the painter wants to paint that triangle from a paint can coverage perspective.

Questions and Discussions

• The instructor opened the floor for student questions and clarifications on both the law of cosines and area formulas for triangles.

• Reminder to complete assignments, prepare for the exam, and note no lab assignments due this week.