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Inverse Trig Functions redux. Some review today, Followed by use and abuse, my favorite (6.6). SAT #1. Easy. SAT #2. Not quite so easy, but still straightforward. SAT #3. A thinker. Review Inverse Trig Functions. What do we already know about the inverse trig functions? List it here. - PowerPoint PPT Presentation
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Review Inverse Trig Functions
What do we already know about the inverse trig functions? List it here.
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?
Review Inverse Trig Functions
What do the graphs of the inverse trig functions look like? Sketch them here.
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
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
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.
Key Point
It can also lead to unexpected answers.
Find the following.
6
7tantan
3
4coscos
3
2sinsin
1
1
1
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
Several problems to work with inverse trig functions
Try these.
__________)5.0(coscos
___________4
3tansin
___________2
3tantan
____________)arctan(tan
1
1
1
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
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
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
Using multiple angle formulas
Find the exact value.
Does it matter what the angles are?
___________5
4arccos
2
1arctansin
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
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
Identities
Verify the identity. Rather than a totally algebraic verification, see how you can simply reason through it. Consider a diagram.
2cossin 11
xx
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
Equation with a twist
Use inverse trig functions and some fancy algebra to solve this equation.
03sin7sin3 2 tt
Equation with a twist
Use inverse trig functions and some fancy algebra to solve this equation.
Embedded quadratic– is it factorable?
03sin7sin3 2 tt
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
Equation with a twist
6
137
32
334497
0373
03sin7sin3
2
2
x
x
xx
tt
Substitute, then use the formula.
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
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