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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 CircuiarFunctionsFindlngvaluesotCircuiarFunctions'
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 xaxis. 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)
“S1 . 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 wanmum 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
LlneaSpeadAngularSpeed 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 menunun ...
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 Fall '11
 PamelaSatterfield

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