7.5 Product-to-Sum and Sum-to-Product Formulas

7.5 Product-to-Sum and Sum-to-Product Formulas

This section is one of the more basic trigonometric concepts, changing products to sums or sums to products.

The proof of the product-to-sum formulas is:

                     sin (u + v) = sin u cos v + cos u sin v
                     sin (u + v) = sin u cos v - cos u sin v
sin (u + v) + sin (u - v) = 2 sin u cos v

This allows us to use these formulas:

(1) sin u cos v = 1/2[sin (u + v) (u - v)]
(2) cos u sin v = 1/2[sin (u + v) (u - v)]
(3) cos u cos v = 1/2[cos (u + v) (u - v)]
(4) sin u sin v = 1/2[cos (u - v) - cos (u + v)]


The proof of the sum-to-product formulas is:

                                           1) u + v = a       and        u - v = b
2) (u + v) + (u - v) = a + b                                                      2) (u + v) - (u - v) = a - b              
3) u = a + b                                                                              3) v = a - b
               2                                                                                               2

         Substitute for u + v and u - v on the right-hand sides of the product-to-sum formulas and for u and v on the left-hand sides. Multiply by 2 and we obtain the following sum-to-product formulas.

(1) sin a + sin b = 2 sin a + b cos a - b

                                           2             2

(2) sin a - sin b = 2 cos a + b sin a - b

                                          2            2

(3) cos a + cos b = 2 cos a + b cos a - b

                                             2             2

(4) cos a - cos b = -2 sin a + b sin a - b

                                            2             2

There are many tremendous youtube videos on this topic. I checked.

Peace.Love.Thad

-Joey

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

6.3 Graphs, Periods, and Even and Odd Functions

Why hello there! This is Nonny the Nonagon stopping by to help everyone with section 6.3. His good friend, Sergei the Circle has decided to come with him to help. Sergei is really good with explaining the graph of sin.


Now, Mr. Wilhelm told us how to look at a graph of sin. Although he had a fancy website, here's a picture showing how a point on a circle coincide with the sin of a point.  (This was Sergei's favorite example)

When you keep going around a circle to show the graph on sin, that wave, formally called a period, will keep repeating. A period is the smallest part of the wavelength on the graph before it repeats. 

Mr. Wilhelm also told us these two equations which go along with the graph below when:

                                                                                                                              

1)                         2) 

Basically, he's just showing us how the two angles are both solutions to the equation, although they are: in completely different quadrants and two different standard position angle measures.



Flashback to Honors Algebra 2 A, we went over functions today in class. Though we were a little rusty on them, we ended up figuring out how this section relates back. But before we get into it, lets review what even and odd functions are.

                                                                   
                                                   

Now, is there any difference between those functions and these?

   
The answer to that would be a no. All we did is change the x to a theta so we can use these equations for circles. But which trigonometric function goes with which? Good question.

Just remember than when you have the two functions below, your answer will always be an odd function.

That's about it for my part of the lesson, so until next time, Bye!

                                                                 

http://www.youtube.com/watch?v=x5ohtlewREI
http://www.youtube.com/watch?v=3nG1ckY7thw

-Kristy Griswold