8.2 The Law of Cosines

Greetings! It's Eleni.
Today, or yesterday, or whenever you choose to learn it, our class learned about the law of cosines.
Here's the link of a math teacher's lesson that Mr. Wilhelm gave us.
http://www.youtube.com/watch?v=24Ibcl3mBQU
This is a picture of the proof:

The a, b, and c are side lengths, while capital letters are angles.  Don't get caught up on which is a, b, or c; here's an easier way to think of it :
Side squared= the sum of the other two sides squared minus two times the same two sides and cos of the angle across from the initial side.
It may sound confusing now...but it actually helps and is better than having to memorize the formulas.

LAW OF COSINES 

When solving a triangle, the law of cosines can be used when you know SAS or SSS. Remember: the law of sines will also come in handy when doing these problems.
SAS
Example problem:
a=5.0, c=8.0, and B=77 degrees




1. draw a triangle and label

2. Since B is the angle between sides a and c, it is easiest to figure out b first.


law of cosines

2. Now that you know a side and it's angle (B and b) you can use the law of sines!

3. Lastly, knowing two out of the three angles of the triangle, we can easily find C.
180-77-35= 68 degrees =C 
SSS
Okay, so doing that last problem step-by-step was a lot of work soooo..... I'm just gonna give you the gist of it.
Example problem:
a=90 b=70 c=40

1. FIND THE LARGEST ANGLE FIRST
even if you have to find the smallest to answer the question, find the largest first.
* the largest angle is across from the largest side (angle A)

2. Now you can use the law of sines to find another angle

3. Subtract both angles from 180 to get the last angle

Another thing we learned in 8.2 was

Heron's Formula

This formula  finds the area of a triangle, any triangle.

S is one half the perimeter
a, b, and c are side lengths.
Another way to find the area of a triangle that's a lot more useful is

---where a and b are any two sides and y is the angle between them (SAS).
Also in the homework, they ask you to combine both of these concepts.
Example: Approximate the area of triangle ABC
A=35.7degrees
C=105.2 degrees
b=17.2
simply use the SAS version of the law of cosines
then use the law of sines like we did in the example above
after, use the a,b, and c side lengths to find the perimeter and therefore s
plug em in the equation.


sooo as we all know mr.w is returning in two days so I thought I'd map it out how far away from us he is.

(At least I think he's in San Diego)
According to google maps, that's only 2340 miles away
and by walking, it should only be roughly a 30 day journey
I hope wilhelm is a fast walker


That's all on 8.2 Bye!
---Eleni

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