MTH 133: Plane Trigonometry

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MTH 133: Plane Trigonometry Radian Measure, Arc Length, and Area Angular and Linear Velocity Thomas W. Judson Department of Mathematics & Statistics Stephen F. Austin State University Fall 2017 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 1 / 20 Contents I 1 2 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 2 / 20

Course Orientation Overview Everything that you need to know about MTH 133 can be found at http://faculty.sfasu.edu/judsontw/math133/index.html Our graphing calculator of choice is Desmos (https://www.desmos.com). The homework system for MTH 133 is WeBWorK (https://webwork.sfasu.edu/webwork2/mth133-f17/). Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 3 / 20 Course Orientation Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 4 / 20

The Unit Circle and Radians Definition 1 The unit circle is the circle of radius one centered at the origin, or x 2 + y 2 = 1. 2 The radian measure of an angle α in the standard position is the directed length of the intercepted arc on the unit circle. Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 5 / 20 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 6 / 20

Radians to Degrees and Degrees to Radians Fact Conversion from radians to degrees is based on 180 degrees = π radians. Convert each of the following from degrees to radians. 1 45 2 90 3 30 4 60 5 120 6 180 7 150 8 225 9 48 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 7 / 20 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 8 / 20

Arc Length of a Circle Fact The length s of an arc intercepted by a central angle of α radians on a circle of radius r is given by s = αr. 1 What is the length of an arc intercepted by α = π/4 on a circle of radius 12 feet? 2 What is the length of an arc intercepted by α = 60 on a circle of radius 2 meters? Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 9 / 20 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 10 / 20

Area of a Sector of a Circle Fact The area A of a sector with a central angle of α radians on a circle of radius r is given by A = αr 2 2. 1 What is the area of a sector of a circle with central angle by α = π/4 and radius 12 feet? 2 What is the area of a sector of a circle with central angle by 60 and radius 2 meters? Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 11 / 20 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 12 / 20

Arc Length and Sector Area Problems 1 Salem, Oregon lies approximately on the 45th parallel. If the radius of the earth is 3950 miles, how far is Salem from the North Pole? 2 A central-pivot irrigation system is watering a circular field of radius 150 feet. The system rotates π/6 radians in one hour. What is the area that is watered in one hour? 3 A bicycle has a chain that connects a chainring of radius 6 inches to a cog of radius 2 inches. If the chainring rotates through an angle of of 20, then how many degrees does the cog rotate? Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 13 / 20 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 14 / 20

Angular Velocity Definition If a point is in motion on a circle through and angle of α radians in time t, then its angular velocity ω is given by ω = α t. Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 15 / 20 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 16 / 20

Linear Velocity Definition If a point is in motion on a circle through and angle of α radians in time t, then its linear velocity ω is given by v = s t, where s is the arc length determined by s = αr. Fact If v is the linear velocity of a point on a circle of radius r, and ω is its angula velocity, then v = rω. Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 17 / 20 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 18 / 20

Angular and Linear Velocity Problems Suppose that a vinyl record is 6.5 inches in diameter and rotates at 45 revolutions per minute. What is its angular velocity and the linear velocity of a point on the edge of the record? The radius of the earth is approximately 3950 miles. If you are standing on the equator, what is your angular velocity? What is your linear velocity? The blade on a table saw rotates at 3450 revolutions per minute. How much faster (in ft/sec) does a 12-in diameter blade strike a piece of wood than a 10 in diameter blade? Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 19 / 20 Plane Trigonometry (MTH 133) Plane Trigonometry Notes Fall 2017 20 / 20

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