Precalculus Trigonometric FunctionsThe Unit Circle 2015

Precalculus WU 10/14

Find one positive and one negative angle co-terminal with the given angle.

5

12

Objectives:

•Find trig function values for special angles using the unit circle.

•Evaluate Trig functions using the unit circle.

•Use domain and period to evaluate trig functions.

•Solve application problems using the unit circle.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

4

The six trigonometric functions of a right triangle, with an

acute angle , are defined by ratios of two sides of the triangle.

The sides of the right triangle are:

the side opposite the acute angle ,

the side adjacent to the acute angle , and the hypotenuse of the right triangle.

The trigonometric functions are

sine, cosine, tangent, cotangent, secant, and cosecant.

opp

adj

hyp

θ

sin = cos = tan =

csc = sec = cot =

opphyp

adj

hyp

hypadj

adj

opp

oppadj

opp

hyp

5

Trigonometric Functions

sin yr

cos xr

tan , 0y xx

Let be an angle in standard position with (x, y), a point on

the terminal side of and r = 2 2 0.x y

csc , 0r yy

sec , 0r xx

cot , 0x yy

y

x

r

(x, y)

6

Example:

2 55

n45

si 6yr

45s

5o 3 5c x

r

6an 23

t yx

Determine the exact values of the six trigonometric functions of the angle .

6sc

245 5c r

y

4ec 53

5s rx

cot 0.536

xy

2 2 2 2

( )

43

3

6 5

6

9

r x y

y

x

(3, 6)

So, we know that trigonometric function values are side length relationships of right triangles.

We can easily evaluate the exact values of trigonometric functions for special angles.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

8

60○ 60○

Consider an equilateral triangle with each side of length 2.

The perpendicular bisector of the base bisects the opposite angle.

The three sides are equal, so the angles are equal; each is 60.

Geometry of the 30-60-90 triangle

2 2

21 1

30○ 30○

3

Use the Pythagorean Theorem to find the length of the altitude, . 3

Special right triangle relationships

1

6045

3 22

1

1

Now, let’s apply it to the unit circle…

What does “unit circle” really mean?

It’s a circle with a radius of 1 unit.

What is the equation of the “unit circle”?

122 yx

, 180 0, 02, 360

3

2

2

1,0

0,1

-1,0

0, -1

, 180 0, 02, 360

3

2

2

Let’s begin with an easy family…4

2

2

2

2

1

45

2

2,

2

2

What are the coordinates?

4

Now, reflect the triangle to the second quadrant…

What are the coordinates?

, 180 0, 02, 360

3

2

2

2

2

2

2

1

45

2

2,

2

2

4

Now, reflect the triangle to the third quadrant…

1

-2

2

2

2

-2

2,

2

2 3

4

What are the coordinates?

Now, reflect the triangle to the fourth quadrant…

, 180 0, 02, 360

3

2

2

2

2

2

2

1

45

2

2,

2

2

4

1

-2

2

2

2

-2

2,

2

2 3

4

-2

2, -

2

2 5

4

What are the coordinates?

, 180 0, 02, 360

3

2

2

2

2

2

2

1

45

2

2,

2

2

4

1

-2

2

2

2

-2

2,

2

2 3

4

-2

2, -

2

2 5

42

2, -

2

2 7

4

30

1 1

2

3

2

6

3

2,1

2

Now, reflect the triangle to the second quadrant.

Complete the family… .6

30

1 1

2

3

2

6

3

2,1

2

Now, reflect the triangle to the third quadrant.

5

6

1

2

-3

2

-3

2,1

2

30

1 1

2

3

2

6

3

2,1

2

Now, reflect the triangle to the fourth quadrant.

5

6

1

2

-3

2

-3

2,1

2

-3

2, -

1

2

What are the coordinates? 7

6

What are the coordinates?

30

1 1

2

3

2

6

3

2,1

2 5

6

1

2

-3

2

-3

2,1

2

-3

2, -

1

2 7

6

3

2, -

1

2

11

6

Let’s look at another “family”3

1

, 180 0, 02, 360

3

2

2

60

3

2

1

2

1

2,

3

2

3

Now, reflect the triangle to the second quadrant

1

, 180 0, 02, 360

3

2

2

60

3

2

1

2

1

2,

3

2

3

Now, reflect the triangle to the third quadrant

1

-1

2

What are the coordinates?

-1

2,

3

2 2

3

3

2

Now, reflect the triangle to the fourth quadrant

1

, 180 0, 02, 360

3

2

2

60

3

2

1

2

1

2,

3

2

3

13

2

-1

2

-1

2,

3

2 2

3

What are the coordinates?

-1

2, -

3

2 4

3

1

, 180 0, 02, 360

3

2

2

60

3

2

1

2

1

2,

3

2

3

13

2

-1

2

-1

2,

3

2 2

3

-1

2, -

3

2 4

3

1

2, -

3

2

What are the coordinates?

5

3

24

x

y (0, 1)

90°2

(–1, 0)

180°

(0, –1)

270°32

360°2 (1, 0)0° 0

Ordered pairs of special angles around the Unit Circle

60° 3

31 ,2 2

45°4

2 2,2 2

30° 6

3 1,2 2

330°116

3 1,2 2

315°74

2 2,2 2

300°53

31 ,2 2

23

120°

31 ,2 2

34

135°

2 2,2 2

56

150°

3 1,2 2

210°76

3 1,2 2

225°5

4

2 2,2 2

240°43

31 ,2 2

sin1y

cos1x

Since r = 1…

cos ,sin ,x y

• Important point:

Since r = 1… sin1y cos

1x

,x y cos ,sin

122 yxBecause ordered pairs around the unit circle (x, y) represent sine and cosine, and the equation of the circle is ,

We have the following identity: 2 2cos sin 1

What if the radius is not 1?

30

30

61

Trigonometric values are functions of the angle – ratios of sides of similar triangles remain the same. So it always holds that .2 2cos sin 1

27

x

y (0, 1)

90°2

(–1, 0)

180°

(0, –1)

270°32

360°2 (1, 0)0° 0

Trigonometric Values of Special Angles

60° 3

31 ,2 2

45°4

2 2,2 2

30° 6

3 1,2 2

330°116

3 1,2 2

315°74

2 2,2 2

300°53

31 ,2 2

23

120°

31 ,2 2

34

135°

2 2,2 2

56

150°

3 1,2 2

210°76

3 1,2 2

225°5

4

2 2,2 2

240°43

31 ,2 2

28

DomainThe domain of the sine and cosine function is the set of all real numbers.

(1, 0)(–1, 0)

(0, –1)

(0, 1)

x

y

x

1 1x

1 1y

Unit Circle

RangeThe point (x, y) is on the unit circle, therefore the range of the sine and cosine function is between – 1 and 1 inclusive.

29

A function f is periodic if there is a positive real number c such that f (t + c) = f (t)

for all t in the domain of f. The least number c for which f is periodic is called the period of f.

x

y

x

Unit Circle

t = 0, 2, …Periodic Function

( ) sinf t t

Period

30

Example:Evaluate sin 5 using its period.

5 - 2 - 2 = sin 5 = sin = 0

Adding 2 to each value of t in the interval [0, 2] completes another revolution around the unit circle.

sin( 2 ) sint n t cos( 2 ) cost n t

( ) sinf t t

x

y

x(–1, 0)

You Try:

a) Evaluate sin

b)Evaluate cos10

3

9

4

Can you?

5tan

6

3sec

4

Evaluate each of the following. Exact values only please.

Even and Odd Trig FunctionsRemember:

if f(-t) = f(t) the function is evenif f(-t) = - f(t) the function is odd

The cosine and secant functions are EVEN.

cos(-t)=cos t sec(-t)=sec t

The sine, cosecant, tangent, and cotangent functions are ODD.

sin(-t)= -sin t csc(-t)= -csc t

tan(-t)= -tan t cot(-t)= -cot t

(1, 0)(–1, 0)

(0,–1)

(0,1)

x

y

x

34

Example:Evaluate the six trigonometric functions at = .

sin 0y

1cos x 0an 01

t yx

1 is un1 defined.csc0y

1 1 11

secx

1 is uc ndefine .o

0t dx

y

(1, 0)(–1, 0)

(0, –1)

(0, 1)

x

y

x

Application:

A ladder 20 feet long leans against the side of a house.The angle of elevation of the ladder is 60 degrees. Find the height from the top of the ladder to the ground.

Application:

An airplane flies at an altitude of 6 miles toward a pointdirectly over an observer. If the angle of elevationfrom the observer to the plane is 45 degrees, find the horizontal distance between the observer and the plane.

.

Homework

4.2 pg. 264 1-51 odd

Trig Races

csc7

22sin

3

cos43

tan 210

csc11

4cos

3

7sin

3

3sin

2

4sec

3

3csc

4

7tan

3

8cot

3

1

2

3

3

2

undef.

3

2

1

2

3

2

1

2

2

3

3

3

HWQ 10/15

5tan

6

csc73sec

4

22sin

3

Evaluate each of the following. Exact values only please.