Inverse Trig Functions redux

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.


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?


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


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


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



Embedded quadratic– is it factorable?

03sin7sin3 2 tt



Embedded quadratic– is it factorable?

No. Go to the quadratic formula.

03sin7sin3 2 tt


6

137

32

334497

0373

03sin7sin3

2

2

x

x

xx

tt

Substitute, then use the formula.


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


6013.

)5657.(sin

6

137sin

6

137sin

1

1

t

t

t


What is the general solution?

)768.1(sin

6

137sin

1

1

t


6013.

)5657.(sin

6

137sin

6

137sin

1

1

t

t

t


What is the general solution? -.6013±2πn,3.7429±2πn

)768.1(sin

6

137sin

1

1

t

Inverse Trig Functions redux

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