Session 15

Agenda:• Questions from 8.1-8.2?• 8.3 – Trigonometric Identities• 8.4 – Graphs of Trigonometric Functions• 8.5 – Inverse Trigonometric Functions• Things to do before our next meeting.

Questions?

8.3 – Trigonometric Identities

• Summary of Trig Identities Already Seen:2 2sin cos 1x x

sin( )tan( )

cos( )

xx

x

1sec( )

cos( )

1csc( )

sin( )

1 cos( )cot( )

tan( ) sin( )

xx

xx

xx

x x

Simplify the following and write as a single trig function.

2

tan( )

sec( )sin ( )

x

x x

• When simplifying trig functions, other techniques such as finding common denominators and factoring may still need to be used.

• Simplify the following to a single trig function3 2

2

cos ( ) cos( )sin ( )

csc( ) csc( )cos ( )

x x x

x x x

Simplify the following as much as possible. Express your answer using only sines and cosines

2

2

sin ( ) cos( )

sin( ) cos( ) cos ( ) sin( ) cos( )

x x

x x x x x

• The following identities are obtained by dividing the fundamental identity by either or .

2 2sin ( ) cos ( ) 1x x 2sin ( )x

2cos ( )x

2 2

2 2

2 2 2

2 2

sin ( ) cos ( ) 1

sin ( ) cos ( ) 1

sin ( ) sin ( ) sin ( )

1 cot ( ) csc ( )

x x

x x

x x x

x x

2 2

2 2

2 2 2

2 2

sin ( ) cos ( ) 1

sin ( ) cos ( ) 1

cos ( ) cos ( ) cos ( )

tan ( ) 1 sec ( )

x x

x x

x x x

x x

Simplify the following completely. (If necessary, assume x is an angle in Quadrant I.) Express your answer using sines and cosines.

For any x, find the value of:

2

2

4cot 4

sin sec sin

x

x x x

2 2

2 2

2sec 2 tan

3cot 3csc

x x

x x

Double Angle Identities

• Use a double angle identity to simplify the following to a an expression involving a single trig function.

sin(2 ) 2sin( )cos( )x x x 2 2

2

2

cos(2 ) cos ( ) sin ( )

2cos ( ) 1

1 2sin ( )

x x x

x

x

3cos( ) cos ( )

sin(2 )cos( )

x x

x x

• Given that and that x lies in Quadrant II, find the

value of .

7csc( )

6x

cos(2 )x

• Given that θ is in Quadrant I and that , express

in terms of x.

• Express in terms of x.

cos( ) x sin( )

sin(2 )

8.4 – Graphs of Trigonometric Functions

• Domain: (-∞, ∞)

• Range: [-1, 1]

• Due to the nature of the unit circle, we know that many angles have the same sine value. Thus, the graph of the sine function repeats itself over and over. We refer to such functions as periodic. The period of the function is the length of the smallest interval that is repeated over and over to form the graph. For sine, the period is 2π.

( ) sin( )f x x

• Domain: (-∞, ∞)

• Range: [-1, 1]

• Period: 2π

( ) cos( )f x x

( ) tan( )f x x

• Vertical Asymptotes:

where k is any odd integer.

• Range: (-∞, ∞)

• Period: π

2

kx

The graphs of the other three trig functions can be remembered by using the fact that each is a reciprocal of sine, cosine, or tangent.

• For example, we know that . The graph of

cosecant is shown below along with the graph of sine.

• Vertical Asymptotes:

where k is any integer.

• Range:

• Period: 2π

1csc( )

sin( )x

x

( ) csc( )f x x

x k

( , 1] [1, )

• We know that . The graph of secant is shown

below along with the graph of cosine.

• Vertical Asymptotes:

where k is any odd integer.

• Range:

• Period: 2π

1sec( )

cos( )x

x

( ) sec( )f x x

2

kx

( , 1] [1, )

• We know that .

• Vertical Asymptotes:

where k is any integer.

• Range: (-∞, ∞)

• Period: π

1cot( )

tan( )x

x

( ) cot( )f x x

x k

Transformations of Trigonometric Graphs

• All the transformations we have applied previously still hold for trigonometric functions.

• Examples:

-- Shift the graph of cosine to the left 3 and then vertically stretch by a factor of 3.

-- Shift the graph of sine to the right 2 and then reflect the graph across the x-axis.

-- Reflect the graph of tangent across the y-axis and then shift the graph up 4 units.

( ) 3cos( 3)f x x

( ) sin( 2)g x x

( ) tan( ) 4h x x

Changing the Period

• We can also change the period of a trigonometric graph, which essentially means stretching or shrinking the graph horizontally.

• Consider the function . First note that this is NOT equal to . The k inside the function changes the period of the graph to .

• The graph to the right is the graph of . The period of this new graph is . The graph has been “shrunk” horizontally.

( ) sin( )f x kx

sin( )k x2

k

( ) sin(2 )f x x2

2

• For a function of the form , the same is true.

• Consider the function .

• The period of the function is

The graph was stretched horizontally.

• The final step is to stretch the graph vertically by a factor of 2 so that its range is now [-2, 2].

( ) cos( )f x kx1

( ) 2cos2

f x x

24

12

• Since the graph of tangent has a period of π, then a function of the form has a period of .

• Consider the function .

• The first thing that is done is to change the period to , essentially shrinking the graph horizontally.

( ) tan( )f x kxk

( ) tan 48

f x x

4

The second thing done is to The last thing to be done shift the graph to the right . is to reflect the graph

across the x-axis.

8

( ) tan 48

f x x

• If a function is written in a form such as ,

you must first factor out the 3 to obtain .

This would give you a period of and a shift left of .

( ) cos(3 )f x x

( ) cos 33

f x x

2

3

3

Sketch a graph (or one period) of .

2( ) 3sin

3 3f x x

Sketch a graph (or one period) of .

6 4( ) 2cos 1

5 5f x x

8.5 – Inverse Trigonometric Functions

• Although sine, cosine, and tangent are NOT one-to-one functions, we can restrict the domains of each of them to obtain a one-to-one function that still encompasses the whole range of each function. The restricted domains are:

• Now that we have restricted each to a one-to-one function, we can define an inverse function for each.

,2 2

,2 2

[0, ]

Inverse Sine Function

• The notation for the inverse sine function is

• The function is defined by:

• Essentially, this means that is the ANGLE whose sine is x.

• The graph of the is shown.

• Domain: [-1, 1] (was range of sine)

• Range: (was restricted domain of sine)

• IMPORTANT: Although there are many angles which have the same sine value, by definition, the angle chosen for the arcsine function must be in the interval , i.e., Quadrants I and IV on the unit circle.

1( ) arcsin( ) sin ( )f x x OR x

1sin ( )x

1( ) sin ( )f x x

,2 2

,2 2

1sin ( ) sin( )y x y x

Evaluate the following.

1

1

1arcsin

2

3sin

2

arcsin(3)

sin sin6

3arcsin sin

4

Inverse Cosine Function

• The notation for the inverse cosine function is

• The function is defined by:

• Essentially, this means that is the ANGLE whose cosine is x.

• The graph of the is shown.

• Domain: [-1, 1]

• Range:

• IMPORTANT: Although there are many angles which have the same cosine value, by definition, the angle chosen for the arccosine function must be in the interval , i.e., Quadrants I and II on the unit circle.

1( ) arccos( ) cos ( )f x x OR x

1cos ( ) cos( )y x y x 1cos ( )x

1( ) cos ( )f x x

[0, ]

0,

Evaluate the following.

1

1

1arccos

2

3cos

2

arccos( )

cos cos3

7arccos cos

6

Inverse Tangent Function

• The notation for the inverse tangent function is

• The function is defined by:

• Essentially, this means that is the ANGLE whose tangent is x.

• The graph of the is shown.

• Domain: (-∞, ∞)

• Range: ; Hor. Asymptotes:

• IMPORTANT: Although there are many angles which have the same tangent value, by definition, the angle chosen for the arctangent function must be in the interval , i.e., Quadrants I and IV on the unit circle.

1( ) arctan( ) tan ( )f x x OR x

1tan ( ) tan( )y x y x 1tan ( )x

1( ) tan ( )f x x

,2 2

,2 2

2y

Evaluate the following.

1

1

arctan 1

1tan

3

4tan tan

3

5arctan tan

6

Inverse Cosecant and Secant Functions

• For arccsc(x) and arcsec(x), you can rewrite the function in terms of arcsin(x) and arccos(x).

• Examples:

1 1csc (2) sin ( )

1 12sec cos ( )

3

1 1 1 11 1csc ( ) sin sec ( ) cosx x

x x

Inverse Cotangent Function

• The graph of cot(x) (with restricted domain in red) and arccot(x) are shown below.

Domain: (0, π) Domain: (-∞, ∞)

Range: (-∞, ∞) Range: (0, π) [Quad. I and II

on unit circle.]

• Evaluate . 1 1

cot3

• Express the following as an algebraic expression involving only x. [Hint: First draw a right triangle where an acute angle has a sine of x.]

• Are there any domain restrictions on x?

tan(arcsin( ))x

Things to Do Before Next Meeting

• Work on Sections 8.3-8.5 until you get all green bars!!

• Write down any questions you have.• Continue working on mastering 8.1-8.2.