8-3 Circular Functions (Presentation)

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8-3 Circular Functions

Unit 8 Trigonometric and Circular Functions



Concepts and Objectives

Unit Circle and Circular Functions (Obj. #28)

Use the unit circle to define values for trig functions.

Determine the measure of an angle based on the

coordinates of its trig value. Determine linear and angular speed of a rotating

point.



Unit Circle

0 1,0

¨ ¸T

© ¹ª º

1

,6

¨ ¸T© ¹ª º

,4

¨ ¸T© ¹ª º

1, T

0,1¨ ¸T

© ¹ª º

1,

¨ ¸T© ¹

ª º,

4

¨ ¸T

© ¹ª º

5 1

,6

T 1,0

¨ ¸T © ¹

ª º

7 1,

6

¨ ¸T © ¹

ª º

5,

4

¨ ¸T © ¹

ª º

4 1, T

0, 1

¨ ¸T© ¹

ª º

5 1,

¨ ¸T© ¹

ª º

7,

4

¨ ¸T© ¹

ª º

11 1,

6

yy

yy

y

y

y

y

yy

yy

y

y

y

y



Circular Functions

The circular functions of real numbers correspond to the

trigonometric functions of angles measured in radians.

r

s = U

x

y

U

(cos s, sin s) = (x, y)

The trig functions and the

circular functions lead to the

same function values,provided we think of the

angles as being in radian

measure.



Circular Functions

Example: Find the exact values of and

cos s = x , so the x -coordinate at

, and at , the coordinates are

T7c s

T¨ ¸© ¹ª º

tan3

T!

7c s

!tany

sx

T5 ¨ ¸© ¹ª º

1,

1

y

x ! 3

1! 3 or z ! !g

3 3 23

2 2 2



Approximating Circular Functions

Example: Find a calculator approximation for each

circular function value.

(a) cos .8 (b) cot .32 9 (c) sec(2.9234)




Example: Find a calculator approximation for each

circular function value.

(a) cos .8 (b) cot .32 9 (c) sec(2.9234)

Make sure your calculator is in radians mode!

(a) cos .8 .2

(b) cot .32 9 = (tan .32 9) .2 2

(c) sec(2.9234) = (cos(2.9234)) . 243




Example: Approximate the value of s in the interval

if cos s = .9 8 .

T« »¬ ¼½

0,2




Example: Approximate the value of s in the interval

if cos s = .9 8 .

cos .9 8 .2Since this value is in the quadrant given, this is our

value.

T« »¬ ¼½

0,2



Approximating Circular Values

Example: Approximate the value of s in if

cos s = .3 .

T« »T¬ ¼½

3,

2



Approximating Circular Values

Example: Approximate the value of s in if

cos s = .3 .

cos .3 .94 .However, in the quadrant given, the cosine should be

negative. Therefore, the answer would be .94 .

T« »T¬ ¼½

3,

2



Exact Circular Values

Example: Find the exact value of s in the interval

if tan s = .

T« »T¬ ¼½

3,



Exact Circular Values

Example: Find the exact value of s in the interval

if tan s = .

tan s = when x = y , which occurs at in the giveninterval.

T« »T¬ ¼½

3,

T5



Linear and Angular Speed

Suppose that point P moves at

a constant speed along a circle

of radius r . The measure of

how fast the position of P is

changing is called linear speed .

If v represents linear speed,

thenr

s

x

y

U

P

!distance

speedtime

!s

v

t




As point P moves along the

circle, ray OP rotates around

the origin. The measure of how

fast �POB is changing is called

angular speed .

Angular speed, symbolized [,

is given as

where U is in radians.

r

s

x

y

U

P

O B

! t

U




Example: Suppose that point P is on a circle with radius

0 cm, and ray OP is rotating with angular speed T/18

radians per second.

(a) Find the angle generated by P in sec.(b) Find the distance traveled by P in sec.

(c) Find the linear speed of P in centimeters per second.






radians per second.

(a) Find the angle generated by P in sec.T[!

18

T U!

18

T TU ! ! ra ia s

18 3






radians per second.

(c) Find the linear speed of P in centimeters per second.

!s

v

t

T!

103

6v

T T! !

10 5cm/sec

18 9



Homework

College Algebr a (brown book)

Page 580: 9-42 (v3s), 54-8 (v3s)

Turn in: 18, 30, 54, 0, , 2, 84

Classwork: Algebr a & Tr igonomet r y (green book)

Page 2 : 15-3 (v3s)

8-3 Circular Functions (Presentation)

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