7.6 Inverse Trig Functions

7.6: Inverse Trigonometric Functions

• Inverse Functions

                                      Where u is the value of a trig function of angle v  

                

              Properties of Inverse Functions

It is important to note that depending on your book,:

• Inverse Sine

or

IMPORTANT! :

                Properties of Inverse Sine

• Inverse Cosine

or

IMPORTANT! :

                   Properties of Inverse Cosine



• Inverse Tangent

or

                Properties of Inverse Tangent

                         y= tan x                                                          y= arctan x

S.O.S!

   When given an angle of inverse sine or inverse tangent, the angle value is restircted to the first and fourth quadrents.  This will help a lot when solving equations.  When given an inverse cosine, the angle is within the first and second quadrents.  Here is an image Mr. Bruns gave us to Help:

 Hope this helps! 6 days till the return of Wilhelm! AHH!
Blessing, EmJ

Section 7.4: Multiple Angle Formulas 

Ok so today in class we learned many more identities and how to derive them.  The formulas that we learned were Double Angle Formulas, Power Reducing formulas, and Half-Angle Formulas.  All of which are identities.

Double Angle Formulas:

We derived all of the double angle formulas from the addition identities.

We know that:

Sin Functions:

                                     

       

This is the Double Angle Formula for Sin functions

Cos Function: 

                              

This equation works but it will only be convenient to use this equation some of the times....in other times it is much more efficient to substitute in other identities that we learned in early sections.  

   

           

You can substitute for sin or cos....

              

One will always be more convenient than another...you just have to find out which one that is so....good luck with that....

Tan Functions:

Power Reducing Formulas:

here we used identities that we just derived...like the ones above this

Sin Functions:

 (subtract 1 and divide by 2)

 

(since Mr.Wilhelm Doesn't like to have negative denominators rewrite it like this)

Cos Functions:

 (add 1 and divide by 2)

Tan Functions:

well we didn't do this in class but it's pretty easy to derive, and at risk of being called a book licker....it's also in there too...all you have to do is put Sin/Cos and you get...

 Now an important thing to note is that these are very rarely used...and a big mistake in using them...is as Mr. Wilhelm put it... using them.

Half Angle Formulas: 

Here we again use formulas and identities that we just learned.

Sin Functions: 

 

Start with the Power reducing formula for sin....and let u=2(theta)

Take the square root of both sides and you get...

 DON'T FORGET THAT IT IS +- THE SQUARE ROOT....but in the finial equation it will be one or the other because weather it is (+) or (-) is based off of the original (sin u/2)

Now we can solve for radians like

because we can change them into special radians.

Cos Functions:

 

Start with the power reducing formula for cos....and let u=2(theta)

take the square root of both sides and you get....

Again don't forget that it is +- the square root but in the end it will be one or the other...unless it is a variable then it can be both

Tan Functions:

here you again just do (sin/cos) and you end up with...

ok so that was pretty much it for today... 

Nonagon 

Mr.Wilhelm we shall miss you these next 5 school days....

k thats it 

-Jennifer Kendall 

Section 7.3: The Addition & Subtraction Formulas

In this section, we derived the six addition and subtraction formulas for sine, cosine, and tangent.

We started by finding the addition identity for sine, beginning with a triangle.

            ∆DFB:                          ∆ABC:                            ∆ABD:

             cos x = DF/DB              sin x = BC/AB              sin y = DB/AD

             DFL DB cos x               BC = AB sin x                DB = ADsin y

                                                                                          cos y = AB/AD

                                                                                            AB = AD cos y

So, we used these findings to figure out the sine of x + y (angle A). To find this, we started with:

Through substitution, we performed the following steps:

We then began to substitute more into the derivation:

By canceling out the AD, we end with:

sin(x+y) = sinycosx + cosxsiny

Here are all six of the addition and subtraction identities:

Sorry for my messy handwriting :)

These are all of the identities, found in similar ways as the sin addition identity. These are true for all values you plug in, because they are identities. 

I think that's about it for 7.3, see you all tomorrow!

- Jessica

                                            

lesson 6.5

To simplify the concept with the sine function, we often use the formula y=a sin [b(x + c)]+d

we can use the same variables for the cosine function y=a cos [b(x + c)]+d

Below are the graphs of y=sinx and y=2sinx

as you can see, the a variable from above designates the amplitude of the graph

these are the graphs of y=sinx and y=sin2x

in this example, you can see than increasing the b variable shortens the period of the graph.

changing the c variable shifts the graph on the x axis a value of -c

this is called phase shift

changing the d variable vertically shifts the graph a distance equal to d

this is called mid-line shift

the graph for cosine behaves very similarly to the sin graph, however the cosine graph reaches its amplitude (1) when x=0. other than this shift, the two graphs are identical. the cosine function is even, while the sine function is odd.