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PreCalc Unit 4 Trig Functions.notebook
1
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))
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