Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)
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Transcript of Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)
![Page 1: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/1.jpg)
Inverse Trig Functions redux
Some review today,Followed by use and abuse, my favorite (6.6)
![Page 2: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/2.jpg)
SAT #1
Easy
![Page 3: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/3.jpg)
SAT #2
Not quite so easy, but still straightforward
![Page 4: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/4.jpg)
SAT #3
A thinker
![Page 5: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/5.jpg)
Review Inverse Trig Functions
What do we already know about the inverse trig functions? List it here.
![Page 6: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/6.jpg)
Review Inverse Trig Functions
Among other things:
All three inverse trig functions are restricted to half a rotation.
Two of the three are continuously increasing. The other continuously decreases.
Want a handout?
![Page 7: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/7.jpg)
Review Inverse Trig Functions
What do the graphs of the inverse trig functions look like? Sketch them here.
![Page 8: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/8.jpg)
Approaches
Either retrieve your chapter 5 handout on inverses or graph on your calculators to fill in these blanks.
_________tan,
________cos,1
________sin,1
1
1
1
xx
xx
xx
![Page 9: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/9.jpg)
Approaches
Either retrieve your chapter 5 handout on inverses or graph on your calculators to fill in these blanks.
2tan,
cos,1
2sin,1
1
1
1
xx
xx
xx
![Page 10: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/10.jpg)
Key Point
A significant characteristic of inverse trig functions is the restriction from the original trig functions.
Remember, this is why we often need to build solutions when solving trig equations.
![Page 11: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/11.jpg)
Key Point
It can also lead to unexpected answers.
Find the following.
6
7tantan
3
4coscos
3
2sinsin
1
1
1
![Page 12: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/12.jpg)
Key Point
It can also give unexpected answers.
Find the following.
Why the difference between input and output values? (This is what item 4 on the handout refers to.)
66
7tantan
3
2
3
4coscos
33
2sinsin
1
1
1
![Page 13: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/13.jpg)
Several problems to work with inverse trig functions
Try these.
__________)5.0(coscos
___________4
3tansin
___________2
3tantan
____________)arctan(tan
1
1
1
![Page 14: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/14.jpg)
Several problems to work with inverse trig functions
Try these.
5.0)5.0(coscos
24
3tansin
2
3tantan
0)arctan(tan
1
1
1
undefined
![Page 15: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/15.jpg)
Diagrams are useful
Rewrite the “arc” part, if it helps, too.Find the exact values.
___________5
3arcsintan
__________13
2arccossin
___________3
2arctansec
___________3
2arctancos
___________3
2arccossin
![Page 16: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/16.jpg)
Diagrams are useful
Rewrite the “arc” part, if it helps, too.Find the exact values.
4
3
5
3arcsintan
13
3
13
2arccossin
3
13
3
2arctansec
13
3
3
2arctancos
3
5
3
2arccossin
![Page 17: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/17.jpg)
Using multiple angle formulas
Find the exact value.
Does it matter what the angles are?
___________5
4arccos
2
1arctansin
![Page 18: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/18.jpg)
Using multiple angle formulas
Think in terms of the Subtraction Formula for Sine. We need only sine and cosine values.
5
1sin
5
2cos
2
1arctan
sincoscossin)sin(
___________5
4arccos
2
1arctansin
u
u
u
vuvuvu
5
3sin
5
4cos
5
4arccos
v
v
v
![Page 19: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/19.jpg)
Using multiple angle formulas
Think in terms of the Subtraction Formula for Sine. We need only sine and cosine values.
25
52
55
2
55
6
55
4
5
3
5
2
5
4
5
1
sincoscossin)sin(
5
4arccos
2
1arctansin
vuvuvu
![Page 20: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/20.jpg)
Identities
Verify the identity. Rather than a totally algebraic verification, see how you can simply reason through it. Consider a diagram.
2cossin 11
xx
![Page 21: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/21.jpg)
Identities
In other words, show that an angle with a sine of x, and an angle with a cosine of x, are complementary.
2cossin 11
xx
![Page 22: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/22.jpg)
Equation with a twist
Use inverse trig functions and some fancy algebra to solve this equation.
03sin7sin3 2 tt
![Page 23: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/23.jpg)
Equation with a twist
Use inverse trig functions and some fancy algebra to solve this equation.
Embedded quadratic– is it factorable?
03sin7sin3 2 tt
![Page 24: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/24.jpg)
Equation with a twist
Use inverse trig functions and some fancy algebra to solve this equation.
Embedded quadratic– is it factorable?
No. Go to the quadratic formula.
03sin7sin3 2 tt
![Page 25: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/25.jpg)
Equation with a twist
6
137
32
334497
0373
03sin7sin3
2
2
x
x
xx
tt
Substitute, then use the formula.
![Page 26: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/26.jpg)
Equation with a twist
6013.
)5657.(sin
6
137sin
6
137sin
1
1
t
t
t
Reverse the substitution. Find a single angle.
)768.1(sin
6
137sin
1
1
t
![Page 27: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/27.jpg)
Equation with a twist
6013.
)5657.(sin
6
137sin
6
137sin
1
1
t
t
t
Reverse the substitution. Find a single angle.
What is the general solution?
)768.1(sin
6
137sin
1
1
t
![Page 28: Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e8f5503460f94b93b7c/html5/thumbnails/28.jpg)
Equation with a twist
6013.
)5657.(sin
6
137sin
6
137sin
1
1
t
t
t
Reverse the substitution. Find a single angle.
What is the general solution? -.6013±2πn,3.7429±2πn
)768.1(sin
6
137sin
1
1
t