Chapter 4 Trigonometric Functions Section 4.1 Angles and Their Measures

Homework Section 4.1 Exercises #17-38, 53, 54

The circumference of a circle is 𝐶=2𝜋𝑟 and where 𝑟=1, 𝐶=2𝜋. The Problem of 360 𝑜 If told that you walked exactly 12 𝑜 , how far did you go? Consider a circle who has a radius of any one unit. The circumference of a circle is 𝐶=2𝜋𝑟 and where 𝑟=1, 𝐶=2𝜋. Therefore 360 𝑜 =2𝜋 or 𝜋= 180 𝑜 . This conversion provides the basis for a linear system of measurements known as radians (abbrv. rad). 𝑟=1

𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠= 180 𝑜 1 𝑟𝑎𝑑𝑖𝑎𝑛=57.30 To convert from degrees to radians multiply by 𝜋 180 𝑜 To convert from radians to degrees multiply by 180 𝑜 𝜋 EX1: Convert each of the following to radians: a. 90 𝑜 90 𝑜 𝜋 180 𝑜 = 𝜋 2 rad b. 30 𝑜 30 𝑜 𝜋 180 𝑜 = 𝜋 6 rad

EX2: Convert the following radian measures into degrees a EX2: Convert the following radian measures into degrees a. 5𝜋 6 rad 5𝜋 6 180 𝜋 = 150 𝑜 b. 15.25 rad 15.25 180 𝜋 = 873.76 𝑜

Arc Length Formula: 𝑠=𝑟𝜃 𝑠 – arc length 𝑟 – radius 𝜃 – central angle in radians EX3: Find the perimeter of a sector whose central angle is 38 𝑜 and radius is 12 meters. 𝑠= 12 38 𝜋 180 𝑠= 38 15 𝜋 or 𝑠=7.96 meters

EX4: The tire on a car has a radius of 20 inches and rotates as a rate of 500 rpm (rotations per minute). Determine the speed of the car in miles per hour. What is one rotation equal to in radians? What is 1 radian equal to in inches? 500𝑟𝑒𝑣 𝑚𝑖𝑛 × 60𝑚𝑖𝑛 ℎ𝑟 × 2𝜋 𝑟𝑎𝑑 1𝑟𝑒𝑣 × 20𝑖𝑛 𝑟𝑎𝑑 × 1𝑓𝑡 12𝑖𝑛 × 1𝑚𝑖 5280𝑓𝑡 ≈59.5 𝑚𝑖𝑙𝑒𝑠 ℎ𝑜𝑢𝑟

Homework Section 4.1 Exercises #17-38, 53, 54