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3.3, 3.4 PPT Handout - 3 Radian Measure and Circular...

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Unformatted text preview: 3 Radian Measure and Circular Functions Trigonometry Circular Functions A unit circle has its center at the origin and a radius of 1 unit. The trigonometric functions of I ‘ ‘ angle 9 in radians are found by ' G choosing a point (x. y) on the unit circle can be rewritten as H.111 functions of the arc length 5. When interpreted this way, they are called circular functions. Unit tilflcxj i” =i want—w 3_3 The Unit Circle and Circular Funcfions CircuiarFunctions-FindlngvaluesotCircuiarFunctions' DeteminingaNurrrberwilhaGivenCirwlarchtionValua - AppifinngwiarFunciions E For any real number 5 represented by a directed arc on the unit circle, sins=y cscs=%(y¢0) cosszx secs=%(x#0) -1 =£ tans- xtx¢0) cots y (yan) The Unit Circle - For a point on the unit circle. its reference are is the shortest arc from the point itself to the nearest point on the x-axis. For example. the quadrant | real number % is associated with the point \ion the unit circle. The Unit Circle Since sin a = yand cos 3 = x, we can replace x and y in the equation of the unit circle ECircular function values of real numbers are [obtained in the same manner as ‘ E trigonometric function values of angles measured in radians. EThis applies both to methods offinding lexact values (such as reference angle I analysis) and to calculator approximations. x2 + y2 =1 to obtain the Pythagorean identity 2 ooszs+sin s=1. i Calculators must be In radian mode g when finding circular function values. FINDING EXACT ClRCULAR FUNCTION VALUES FINDING EXACT CIRCULAR FUNCTION VALUES b Example 2(a) D Example 1 Use the figure to find the exact values of -4_:r 4_:r sun3 andcos3. Find the exact values of sin (-3rr), cos (—370, and tan (—321). APPROXlMATlNG CIRCULAR FUNCTION VALUES (continued) FINDING EXACT CIRCULAR FUNCTION VALUES bExample 3 ) Example 2(b) Find a calculator approximation for each circular function value. Use the figure and the definition of tangent to find the exact value of lan(—9«’5«) = tan(7—’r (C) out 1.3209 = .2552 (d) sec —2.9234 = -1.0243 " “353%??4915 4 4 x ( - 4) “S-1 . 323285532 Caution A common error In trigonometry is using a calculator in degree mode when radian mode should be used. Remember; if you are finding a circular funcfion value of a real number, the calculator must be In radian mode. FiNDING A NUMBER GIVEN ns ’ Exampie 4a)) CIRCULAR FUNCTION VALUE Linear Speed Given a point P that moves at a constant speed along a circle of radius rand center 0. The measure of how fast the position of P is changing is its linear speed. distance or V = g speed = time t v is the linear speed. s is the length of the arc traced b! point P at time t wan-mum FINDING A NUMBER GIVEN ITS ’ Example 4(a) CIRCULAR FUNCTION VALUE Approximate the value of s in the interval if sin 3 = .3210. 3_4 Linear and Angular Speed LlneaSpead-AngularSpeed Angular Speed As point P moves along the circle. ray OP rotates about the origin. The measure of how fast angle P03 is changing is its angular speed. =9 l a) is the angular speed. 9 is the measure of angle P08 (in radians) traced by point P at timet momma—W Angular Speed 0 w:— I (m in radians per unit time, 0 in radians) FINDING ANGULAR SPEED OF A PULLEY ' Example 2 AND LINEAR SPEED OF A BELT Abelt runs a pulley of radius 5 in at 120 revolutions per minute. (3) Find the angular speed of the puiley in radians per second. (b) Find the linear speed of the belt in inches per second. USING LINEAR AND ANGULAR SPEED Exam Ie ’ p 1 FORMULAS Suppose that P is on a circle with radius 151i?” and ray OP is rotating with angular speed Ti radian per second. (a) Find the angle generated by P in 10 seconds. m=~g=>3=wl (b) Find the distance traveled by P along the circle in 10 seconds. s = r6 (c) Find the linear speed of P in centimeters per second. V = E t menu-nun ...
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