PreCalc Unit 4 Trig Functions.notebook

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December 03, 2018

Jan 7­6:35 PM

Glöckner

Remember what an inverse function is?

How do you make one from a graph?

How would you make one for f(x) = sin(x)?

Jul 26­11:55 AM

4.6 Inverse Trig Functions

Lesson goals:

1) I can evaluate and graph inverse trigonometric functions.

2) I can find compositions of trigonometric functions

PreCalc Unit 4 Trig Functions.notebook

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December 03, 2018

Jan 21­10:40 AM

Jan 7­6:35 PM

f(x)=sin(x)

y=x

Take your original function and reflect it over y=x:

PreCalc Unit 4 Trig Functions.notebook

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December 03, 2018

Jan 7­6:35 PM

Reflect over y=x:

f(x)=sin(x)

y=x

Jan 7­6:35 PM

But this new graph isn't a function....why?

f(x)=sin(x)

y=x

PreCalc Unit 4 Trig Functions.notebook

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December 03, 2018

Jan 7­6:35 PM

OK, so we will RESTRICT the domain of the sine graph so the sin­1(x) will look like a function:

f(x)=sin(x)

y=x

sin ­1(x) or arcsine x, can be interpreted as the angle or arc between ±π/2 with a sin of x

Jan 7­6:35 PM

y=xSo we restrict sin(x):

domain = [­π/2, π/2]

range = [­1,1]

Then for sin­1(x):

domain = [­1,1]

range = [­π/2, π/2]

PreCalc Unit 4 Trig Functions.notebook

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December 03, 2018

Aug 7­9:50 PM

GUIDED PRACTICE

A) arcsin ( ) B) arcsin (­1)

C) sin ­1 (­2π) D) sin ­1 3

Jan 7­6:35 PM

f(x)=cos(x)

y=x

OK, Now let's try cos(x):

PreCalc Unit 4 Trig Functions.notebook

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December 03, 2018

Jan 7­6:35 PM

f(x)=cos(x)

y=x

OK, Now let's try cos(x):

Jan 7­6:35 PM

f(x)=cos(x)

y=x

the red graph is not a function though....hmmm

PreCalc Unit 4 Trig Functions.notebook

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December 03, 2018

Jan 7­6:35 PM

f(x)=cos(x)

y=x

so, restrict the domain until it passes the vertical line test:

f '(x)=cos­1(x)

Jan 7­6:35 PM

f(x)=cos(x)

So for cos­1(x) we have

domain = [­1, 1]

range = [0, π]

f '(x)=cos­1(x)

Restrict cos(x) so then

domain = [0, π]

range = [­1, 1]

PreCalc Unit 4 Trig Functions.notebook

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December 03, 2018

Aug 7­9:50 PM

GUIDED PRACTICE

A) arccos( ) B) arccos (­1/2)

C) cos ­1 (­2π) D) cos ­1 2.5

Jan 7­6:35 PM

f(x)=tan(x)

PreCalc Unit 4 Trig Functions.notebook

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December 03, 2018

Jan 7­6:35 PM

f(x)=tan(x)

f '(x)=tan­1(x)

For restricting tan(x)

domain = [­π/2, π/2]

range = (­ ∞, ∞)

So then for tan­1(x)

domain = (­ ∞, ∞)range = [­π/2, π/2]

Aug 7­9:50 PM

GUIDED PRACTICE

A) arctan ( ) B) arctan 0

C) tan ­1 (­1) D) tan ­1 √3

PreCalc Unit 4 Trig Functions.notebook

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December 03, 2018

Aug 7­9:50 PM

Aug 7­9:50 PM

Sketch the graph of each function:

y = arcsin 3x y = tan ­1 2x y = arccos 2x****You can sketch the inverse trig graphs by rewriting the functions like so: sin y = x, cos y = x, tan y = x; then, you make a table, assigning values to y, plot those points and connect them w/a smooth curve****

PreCalc Unit 4 Trig Functions.notebook

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December 03, 2018

Aug 7­9:50 PM

Real Life Example ... (really )Luke has purchased a new flat-screen tv. So that his family will be able to see he has decided to mount it as shown below:

7.5 ft

2.5 ft

a) Write a function modeling the distance d of the maximum viewing angle, θ, for Luke if his eye level when sitting is 3 feet above the ground.

b) Determine the distance that corresponds to the maximum viewing angle.

θ=θ2­θ1

Aug 7­9:50 PM

So, like any inverse function sin­1 and arcsin "undo" the original sin function.

What does that mean?

When we compose them it is like tying and untying your shoes ­ you end up where you started...

PreCalc Unit 4 Trig Functions.notebook

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December 03, 2018

Aug 7­9:50 PM

Examples:1) tan(tan-1(π/3))

2) cos-1(cos(3π/4))

*3) arcsin(sin(2π/3))

**4) arctan(tan(π/2))

Aug 7­9:50 PM

Examples:

5) cos-1(sin(π/3))

Ok, so you have to do a little work when we blend different trig functions....

6) sin(arctan(5/12))