Transcript File
Radian Measure and the Unit Circle 2006 John Wiley and Sons. All rights reserved. Trigonometry by Cynthia Y. Young, © 2009 Chapter 3 2006 John Wiley and Sons. All rights reserved. Trigonometry by Cynthia Y. Young, © 2009 3.1 Radian Measure Radians – another way to measure ___________. Central angle – __________ is at the _____________ of a circle Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Angles Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. 1 Radian = the measure of the _________________ whose _____________________ is equal to the _____________________ The _____________ for the side and the radius must be the __________or the resulting angle measure will be incorrect. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. To Find Radian Measure: Multiply by a form of one: Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Converting Between Degrees and Radians -300⁰ 139⁰10’ Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Convert to radians. Leave the first answer as a multiple of π. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Convert to degrees. 1.74 Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Convert to degrees. Write answers to the nearest minute. We can draw the unit circle with special angles in degrees and radians. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Special Angles in the Unit Circle To evaluate trig functions for angles measured in ________________. Convert ____________________. Recall ______________________ for ___________________(set up special right triangles, with reference angles, if needed). Pay attention to the ____________________. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Trig Functions Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Find the exact value of each expression without using a calculator. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. 2006 John Wiley and Sons. All rights reserved. Trigonometry by Cynthia Y. Young, © 2009 3.2 Applications of Radian Measure Θ MUST BE MEASURED IN _______________. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. To Find Arc Length: Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Find the arc length intercepted by a central angle θ in a circle of radius r. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Find the distance between the cities. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Two gears are adjusted so that the smaller gear drives the larger one, as shown in the figure. If the smaller gear rotates through an angle of 240⁰, through how many degrees does the larger gear rotate? Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. A ____________ of a circle. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Sector of a Circle: Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Calculating Area of a Circular Sector: Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Find the area of the sector. A circular sector has an area of 25 in2. The radius of the circle is 2 in. What is the arc length of the sector? Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: 2006 John Wiley and Sons. All rights reserved. Trigonometry by Cynthia Y. Young, © 2009 3.3 The Unit Circle and Circular Functions Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Unit Circle: Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. The Unit Circle Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Trig Functions on the Unit Circle: Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Find sin s, cos s, and tan s. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Find the exact circular functionn value for each. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Find a calculator approximation for each circular function value. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Evaluate the six circular function values of θ on the unit circle. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Taking the inverse function of a ratio, allows you to find the angle: Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Inverse Functions Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Find the value of s. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Find the exact value of s in the given interval. 2006 John Wiley and Sons. All rights reserved. Trigonometry by Cynthia Y. Young, © 2009 3.4 Linear and Angular Speeds The _______________ an object travels per _____________________: Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Linear Speed A car travels at a constant speed around a circular track with circumference of 3 miles. If the car records a time of 12 minutes for 7 laps, what is the linear speed in miles per hour? Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: The _________________ an object gets through per _____________________: Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Angular Speed Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Relating Linear and Angular Speeds: A) The angle generated by P in time t. B) the distance traveled by P along the circle in time t. C) the linear speed of P. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Find the following for Point P on a circle with radius r rotating at an angular speed. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Find the missing variable in the angular speed formula. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Find the missing variable in the linear speed formula. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved. EX: Find the missing variable. Trigonometry by Cynthia Y. Young, © 2006 John Wiley and Sons. All rights reserved.