4.2/4.34.2/4.34.2/4.34.2/4.3Trigonometric Functions: Trigonometric Functions:

The Unit CircleThe Unit Circle

&&

Right Triangle TrigonometryRight Triangle Trigonometry

Quick Review

Give the value of the angle in degrees.

21.

3

2. 4

Use special triangles to evaluate.

3. cot4

74. cos

6

5. Use a right triangle to find the other five trigonometric

functions of the

4 acute angle given cos

5

Quick Review Solutions

Give the value of the angle in degrees.

21.

3

2. 4

Use special triangles to evaluate.

3. cot 4

74. cos

6

5. Use a right triangle to find the other five tr

120

45

1

igo

3 / 2

nometric

4

sec

functions

5 /

of th

4, sin

e acute angl

3 / 5, csc

e given

5 / 3, tan 3/ 4, cot 4 / 3

cos5

What you’ll learn about• How to identify a Unit Circle and its relationship

to real numbers• How to evaluate Trigonometric Functions using

the unit circle• Periodic Functions of sine and cosine functions… and whyExtending trigonometric functions beyond triangle

ratios opens up a new world of applications to model and solve real-life problems.

Initial Side, Terminal Side

Positive Angle, Negative Angle

Evaluating Trig Functions of a

Nonquadrantal Angle θ 1. Draw the angle θ in standard position, being

careful to place the terminal side in the correct quadrant.

2. Without declaring a scale on either axis, label a point P (other than the origin) on the terminal side of θ.

3. Draw a perpendicular segment from P to the x-axis, determining the reference triangle. If this triangle is one of the triangles whose ratios you know, label the sides accordingly. If it is not, then you will need to use your calculator.

4. Use the sides of the triangle to determine the coordinates of point P, making them positive or negative according to the signs of x and y in that particular quadrant.

5. Use the coordinates of point P and the definitions to determine the six trig functions.

Trigonometric Functions of any Angle

2 2

Let be any angle in standard position and let ( , ) be any point on the

terminal side of the angle (except the origin). Let denote the distance from

( , ) to the origin, i.e., let . Then

si

P x y

r

P x y r x y

n csc ( 0)

cos sec ( 0)

tan ( 0) cot ( 0)

y ry

r y

x rx

r xy x

x yx y

Unit CircleThe unit circle is a circle of radius 1 centered at the origin.

Trigonometric Functions on Unit Circle Let be any real number, and let ( , ) be the point corresponding to

when the number line is wrapped onto the unit circle as described above.

Then

1sin csc ( 0)

cos

t P x y

t

t y t yy

t x

1

sec ( 0)

tan ( 0) cot ( 0)

t xx

y xt x t yx y

The 16-Point Unit Circle

Example Using one Trig Ratio to Find the

Others

Find sin and cos , given tan 4 / 3 and cos 0.

Example Using one Trig Ratio to Find the

Others

Find sin and cos , given tan 4 / 3 and cos 0.

Since tan is positive the terminal side is either in QI or QIII.

The added fact that cos is negative means that the terminal

side is in QIII. Draw a reference triangle with 5, -3,

and -4.

sin -

r x

y

4 /5 and cos -3/5

Periodic Function

A function ( ) is if there is a positive number such that

( ) ( ) for all values of in the domain of . The smallest such

number is called the of the function.

y f t c

f t c f t t f

c

periodic

period