Day 51-Inverse Trigononmetric Functions

8/11/2019 Day 51-Inverse Trigononmetric Functions

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Graphs of TrigonometricFunctions and Inverses


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What you will learn

Evaluate and graph the inverse sine, cosine

and tangent functions.

Evaluate compositions of trigonometric

functions.

2


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3

Plan for the Day

Quick review of graphing

Graphing Inverses

Evaluating Inverses

Compositions

You will need your calculator andunit circle


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Get out your calculator:

Try These

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Get out your calculator:

Try These

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22.6o17.1o

50.0o48.2o


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Inverse Functions


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Inverse Functions

What is the definition of an inverse function?

What kind of test did we use to check for

inverse functions?

Will the inverses of Sine, Cosine and Tangent

be a function?


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Graph of the Cosine Function

To sketch the graph ofy= cosxfirst locate the key points.

These are the maximum points, the minimum points, andthe intercepts.

10-101cosx

0x2

2

3

2

Then, connect the points on the graph with a smooth curve

that extends in both directions beyond the five points. A

single cycle is called a period.y

2

3

2

22

3

2

2

5

1

1

x

y= cosx


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y

2

2

3

2

1 1

x


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Inverse of Cosine (Page 324)On the interval, [0, ] cosine is decreasing

On the interval, [0, ] y = cos x takes on its fullrange of values [-1,1]

On the interval, [0, ] cosine is one-to-oneSo, on this restricted interval, cosine does have

an inverse function written as

y = arccos x or y = cos-1x

The angle whose cosine is xThe domain of y = arccos x is [-1, 1] and the

range is [0, ].


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Evaluate the following

arccos

arccos -1

cos-10

arccos (-)


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Graph of the Sine Function

To sketch the graph ofy= sinxfirst locate the key points.

These are the maximum points, the minimum points, andthe intercepts.

0-1010sinx

0x2

2

32

Then, connect the points on the graph with a smooth curve

that extends in both directions beyond the five points. A

single cycle is called a period.y

2

3

2

22

3

2

2

5

1

1

x

y= sinx


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Inverse of the Sine (sin-1 )

Switch the x and y

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0-1010sinx

0x2

2

3

2

0-1010

sin

-1

x 0

x

2

2

3

2


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Inverse Sine

16

y

2

2

3

2

11

x

2


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Inverse of Sine (page 322)

On the interval, [-/2, /2] sine is increasingOn the interval, [-/2, /2] y = sin x takes on its

full range of values [-1,1]

On the interval, [-/2, /2] sine is one-to-oneSo, on this restricted interval, sine does have an

inverse function writteny = arcsin x or y = sin-1x

The angle whose sine is xThe domain of y = arcsin x is [-1, 1] and the

range is [-/2, /2].


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Evaluate the following

arcsin

arcsin 1

sin-10

arcsin (-)


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y

x

2

3

2

3

2

2

Graph of the Tangent Function

2. range: (, +)

3. period:

4. vertical asymptotes:

nnx 2

1. domain : all realx nnx

2

Properties of y= tanx

period:

To graphy= tanx, use the identity .x

x

x

cos

sintan

At values ofxfor which cosx= 0, the tangent function isundefined and its graph has vertical asymptotes.


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Inverse Tangent

20

y

x

2

3

2

3

2

2


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Inverse of Tangent (page 324)

On the interval, (-/2, /2) tangent is increasing

On the interval, (-/2, /2) y = tan x takes on its fullrange of values (-, )

On the interval, (-/2, /2) tangent is one-to-oneSo, on this restricted interval, tangent does have an

inverse function written

y = arctan x or y = tan-1x

The angle whose tangent is xThe domain of y = arctan x is (-, ) and the range is

(-/2, /2).


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Composite Functions (Page 326)

What is a composite function?

Remember f(f-1(x)) = x f-1 (f(x)) = x then

Think about it what is

arcsin(sin /3) = ??

cos (arccos ) = ??

cos (arcsin ) = ??


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If -1 x 1 and /2 y /2 thensin (arcsin x) = x and arcsin (sin y) = y

If -1 x 1 and 0 y thencos (arccos x) = x and arccos (cos y) = y

If x is a real number and/2< y < /2 thentan (arctan x) = x and arctan (tan y) = y


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Lets look at this statement

If -1 x 1 and /2 y /2 thensin (arcsin x) = x and arcsin (sin y) = y

What if y >/2 and not within the range stated??

arcsin(sin 2/3) = ??


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Remember completing problems like:

Given the cosine of an acute angle is 2/3, find the

tangent.


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Homework 28

Section 4.7, p. 328:

1-25 every other odd, 37-41 odd, 49, 53

Day 51-Inverse Trigononmetric Functions

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