THE UNIT O CIRCLE

Trigonometric Functions of Real Numbers6.3

Mrs. Crespo 2011

The Unit Circle With radius r=1 and a

center at (0,0).

r = 1

(0,0)

(0,1)

(1,0)(-1,0)

(0,-1)

S = arc length

S

r θ =

S

1= = S

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(0,0)

(0,1)

(1,0)(-1,0)

(0,-1)

The Unit Circle To find the terminal

point P(x,y) for a given real number t, move t units on the circle starting at (1,0).

P(x,y)

Move counterclockwise if t > 0.

Move clockwise if t < 0.

-t

t

P(x,y)

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The Unit Circle and the Trig. Functions With radius r=1, then

xr

cos t = x1

= = x

y

rsin t = y

1= = y

y

xtan t =

ry

csc t = 1y

= rx

sec t = 1x

=

x

ycot t =

Mrs. Crespo 2011r =

1

(0,0)

(0,1)

(1,0)(-1,0)

(0,-1)

y

X

Example 1 P(-3/5 ,-4/5) is on the terminal side of t. Find sin t, cos t, and tan t.

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(0,1)

(1,0)(-1,0)

(0,-1)

(+,+)

(+,-)(-,-)

(-,+)

P(-3/5 ,-4/5)

sin t = y = -4/5 -3/5 cos t = x=

y

xtan t = = = 4/3

Your Turn 1 P(4/5 , 3/5) is on the terminal side of t. Find sin t, cos t, and tan t.

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(0,1)

(1,0)(-1,0)

(0,-1)

P(4/5 ,3/5)

sin t = y = 3/5 4/5 cos t = x=

y

xtan t = = = 3/4

Example 2

With P(t)

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Given the following sketch.

(0,1)

(1,0)(-1,0)

(0,-1)

P(t) =(4/5 ,3/5)

t

P(t + π) =(-4/5 ,-3/5)

Example 2

Find P(t + π)π = 180˚

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Given the following sketch.

180˚ forms a straight line

adding π means moving ccw

(0,1)

(1,0)(-1,0)

(0,-1)

P(t) =(4/5 ,3/5)

On QIII (-,-) t

t + π

P(t - π) =(-4/5 ,-3/5)

Example 2

Find P(t - π)π = 180˚

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Given the following sketch.

180˚ forms a straight line

subtracting π means moving cw

(0,1)

(1,0)(-1,0)

(0,-1)

P(t) =(4/5 ,3/5)

Still on QIII (-,-)

t - π

Example 2

Find P(-t)-t means moving cw

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Given the following sketch.

(0,1)

(1,0)(-1,0)

(0,-1)

P(t) =(4/5 ,3/5)

t

Reflect on x-axis means x-axis is the mirror line

Mirror Line Samples

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Example 2

Find P(-t)-t means moving cw

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Given the following sketch.

(0,1)

(1,0)(-1,0)

(0,-1)

P(t) =(4/5 ,3/5)

On QIV (+,-)

t

-t

Reflect on x-axis means x-axis is the mirror line

P(-t) =(4/5 ,-3/5)

Example 2

Find P(-t - π)

from -t move cw

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Given the following sketch.

(0,1)

(1,0)(-1,0)

(0,-1)

P(t) =(4/5 ,3/5)

On QII (-,+)

t

-t

P(-t - π) =(-4/5 ,3/5)

180˚ forms a straight line

subtracting π means moving cw

-t - π

π = 180˚

Your Turn 2

a) P(t + π)

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(0,1)

(1,0)(-1,0)

(0,-1)

Given P(t)=(-8/17 ,15/17) , find:

b) P(t - π)

d) P(-t - π)

c) P(-t)

P(t)=(-8/17 ,15/17)

P(t + π)=(8/17 ,-15/17)

P(t - π)=(8/17 ,-15/17) P(-t)=(-8/17 ,-15/17)

P(-t - π)=(8/17 ,15/17)

The Unit CircleWe know that:

Π = 180˚

(0,0)

(0,1)

(1,0)(-1,0)

(0,-1)

2 Π = 360˚

360˚ is one full rotation.

2ππ

π

2

3π

2Mrs. Crespo 2011

Then, P(x , y) = P(cos t, sin t)

Examples

Find

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=cos π

2 sin π

2

sin 3π

2

P(x , y) = P(cos t, sin t) on the Unit Circle

cos 3π

2

cos π sin π

sin 2πcos 2π

=

=

==

=

==

1

0

-1

0

0

-1

0

1

The Unit Circle(0,1)

(1,0)(-1,0)

(0,-1)

Start with QI.

•The denominators for all coordinates is 2.

•The x-numerators going from 60˚, 45˚ to 30˚, write 1, 2, 3.

•The y-numerators going from 30˚, 45˚ to 60˚, write 1,2,3.

•Square root all numerators.

2π

π 2

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45˚

30˚

60˚

0˚180˚

90˚

120˚

150˚

360˚330˚

300˚

210˚

240˚

270˚

135˚

315˚225˚

π 0

2π3 π

4

7π4

5π33π

2

π3

π6

5π6

3π4

5π4

7π6

4π3

11π6

(-1/2 ,√3/2) (√2/2 , √2/2)

(√3/2 ,1/2)

(-√3/2 ,-1/2) (√3/2 ,-1/2)

(√2/2 , -√2/2) (1/2 ,-√3/2)

(1/2 ,√3/2)

(-√2/2 , √2/2)

(-√3/2 ,1/2)

(-√2/2 , -√2/2) (-1/2 ,-√3/2)

Once QI special angles have points determined, the rests are easy to find out.

Degrees

Points

Radians

Formulas for Negativessin (-t) = - sin (t) cos (-t) = cos (t) tan (-t) = - tan (t)csc (-t) = - csc (t) sec (-t) = sec (t) cot (-t) = - cot (t)

Mrs. Crespo 2011

EXAMPLES

-2

2

-1

-√3

Estimating

sin (0) =

P(x , y) = P(cos θ, sin θ)

sin (1) =

cos (3) =

cos (-6) =

cos (4) =

sin (5) =

cos (0)=

0

sin (3) =

.02 .05

1 1 1

.09

1

Even and Odd Functions

Even Functions Odd Functions

The form is f(-x) = f(x). Signs of both coordinate

points change. Symmetric with respect to

y-axis.

The form is f(-x) = - f(x). Signs of y-coordinates do

not change. Symmetric with respect to

the origin.

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sin (-t) = - sin (t) cos (-t) = cos (t) tan (-t) = - tan (t)csc (-t) = - csc (t) sec (-t) = sec (t) cot (-t) = - cot (t)

TURN TO PAGE 441 AND OBSERVE THE GRAPHS ON THE TABLE.

Homework

Mrs. Crespo 2011

PAGE 444 : 1- 20 ODD

Resources

Textbook: Algebra and Trigonometry with Analytic Geometry by Swokowski and Cole (12th Edition, Thomson Learning, 2008).

http://www.mathlearning.net/learningtools/Flash/unitCircle/unitCircle.html

http://www.mathvids.com/lesson/mathhelp/36-unit-circle www.embeddedmath.com/downloads tutor-usa.com/video/lesson/trigonometry/ 4059-unit-circle. PowerPoint and Lesson Plan customization by Mrs. Crespo 2011. Ladywood High School

Mrs. Crespo 2011