Trigonometric FunctionsSection 1.6

Radian Measure• The radian measure of the angle ACB at the center of

the unit circle equals the length of the arc that ACB cuts from the unit circle.

Radian Measure• An angle of measure θ is placed in standard position

at the center of circle of radius r,

In this course, you will see that so much of calculus only works when measuring angles in radians!!!

Trigonometric Functions of ThetaThe six basic trigonometric functions of are defined as follows:

sine: sin cosecant: csc

cosine: cos secant: sec

tangent: tan cotangent: cot

y rr yx rr xy xx y

(1,0)(–1,0)

(0,–1)

(0,1)

0 0360 2

30 645 4

60 390

2

120

23

135

34

150

56

180

21076

22554 240

43

27032

30053

31574

330 116

1 3,2 2

1 3,2 2

3 1,2 2

3 1,2 2

2 2,2 2

2 2,2 2

3 1,2 2

3 1,2 2

2 2,2 2

2 2,2 2

1 3,2 2

1 3,2 2

Other Assortments…

A function is if there is a positive number such

that for every value of . The smallest value

of is the of .The functions cos , sin , sec and csc are periodic wit

f x p

f x p f x x

p fx x x x

periodic

periodh

period 2 . The functions tan and cot are periodic with period .

x x

• The graphs of cos x and sec x are even functions because their graphs are symmetric about the y-axis.• The graphs of sin x, csc x, tan x and cot x are odd

functions.

Inverse Trigonometric Functions

• None of the six basic trigonometric functions graphed in Figure 1.42 is one-to-one. These functions do not have inverses. However, in each case, the domain can be restricted to produce a new function that does have an inverse. • The domains and ranges of the inverse trigonometric

functions become part of their definitions.

Inverse Trigonometric Functions

1

1

1

1

1

1

cos 1 1 0

sin 1 1 2 2

tan 2 2

sec 1 0 ,2

csc 1 , 02 2

cot 0

y x x y

y x x y

y x x y

y x x y y

y x x y y

y x x y

Function Domain Range

Inverse Trigonometric Functions

A key to remember:Inverse trig functions ARE angles!!!

1sin x “The sine (or othertrig function) ofwhat angle is x?”

How to “read” inverse trig expressions…

Also remember: Inverse sine and inverse tangent are always on the right half of the unit circle, and inverse cosine is always on the top half of the unit circle…

Guided Practice

5sin6

Evaluate each of the following:

12

3cot4

1

13tan6 3

3 csc

3 2 3

3

4cos3

21sec4

2

12

Guided Practice

2sin 4 3y x

Determine (a) the period, (b) the domain, and (c) therange, and (d) draw the graph of the function.

2sin 4 4 3x

Horizontal shift (phase shift) left , horizontal shrinkby 1/4, vertical stretch by 2, vertical shift up 3

4

(a) Period:2 2

4 2b

(b) Domain: ,

(c) Range: 1,5 (d) How does the graph look?

Guided Practice

cos 0.6,x Solve the equation in the specific interval.

0 3x

0.927,5.356,7.210x

3sin ,2

x x

22 , 23 3

x n n

n is any integer

Guided Practice

1 2sin2

Evaluate each of the following:

4

1cos 0.23 1.339

1 1cot cos2

33

cot3

1 9tan sin13

9 0.9592 22

Let’s complete this last one with an analytic graph…