Well this is another last minute blog

Aim:How do we solve trigonometric equations with more than one function?

so.. lets start with what I mean when i write a trigonometric function.
A trigonometric function is anything similar to :
sin, cos, tan, csc, sec and cot. 
A trigonometric equation with more than two functions is a equation that has two of the functions. 

An example : 

Now, that we have established what the aim means we will learn how to solve it.

Lets say we get  a trigonometric equation like : 

Now you find when sin equals 0 and 1. 

We are done. 

Spring break .. is it a break? nope not really :)

Another blog .....
today's blog focuses on the topic of: How do we solve linear trigonometric equations?

to solve a linear trigonometric equation you have to follow these steps:

1.You have to get the trigonometric alone.
2. Use the inverse of the expression to solve for the angle.
3. Check for multiple answers.

To solve these equations we have to use a special circle like the one below.

solve for all the values such that 0 is less than theta and theta is less than 2pi.

 Example:

so now bye :).

Another blog just in time.

The question of the day is ....

 How do we graph trig functions?

So sadly in trigonometry we have many different kinds of graphs.
You might have heard about my friends (and yes I have friends) SIN,COS and TAN. 
As you have battled your way through half of the year with trigonometry, I must tell you it really does not get better. (hehe my evil laugh). Well, back to trigonometry during your journey in trigonometry we have learned about the "beauties" of SIN,COS and TAN.Using their "beauty" we will combine them to learn how to graph those "beauties". 

WHOOPS.. i forgot to mention when graphing SIN, COS and TAN there is a thing called a "period". 
A period is basically the "time" that it takes for the function to make one cycle. 
For Example : 

Besides, the period there is also something called the mid line and   the Amplitude. 

For example :

Finally, we need to know the equation .
The equation will often look like this :

Y=A Sin(bx)+C
A= the Amplitude
Sin could be cos or tan ,in this case its sin. 
B= the period 
C= the mid line
Remember: you find the period by dividing 2pi by the cycle.


Lets start with my friend : SIN
- it ALWAYS starts at the origin. 

So lets say we have an equation like : 
y= 2sin(2x)+0
A= 2
B= 2
C = 0

So this graph would look like this : 

For the amplitude its suppose to be two,pardon my lack of art skills.

Now onto COS: 
- does not start at the origin but starts at the amplitude.

An equation:
Y= 4 cos(4x)+ -1

A= 4 
B= 4
C= -1

Now graph time:

Finally, its TAN. 
- Tan is a special case, it never touches the asymptotes that are each cycle.
- no Amplitude 

First off you need to know how to find the asymptotes
so basically when you find them you 
use  the four angles :
90, 180,270 ,360 . 
you can also use the negative angles. 
so basically you divide it by 360 by those angles and then like mr.schnat says " divide it by 180 and slap a pi on it. to get the asymptotes.

lets say the asymptote is 2, and 4 
then we graph it like this :

Notice it does not have a amplitude and the line are like snakes that go up on one side and down on the other. 
also they never touch the asympotes .
that concludes my blog . yay !

Something for homework

Hi, to you all, well not really just Mr. Schnat
but you know what I mean :)

Today's question is : Why is the name Pythagorean identity appropriate ?

So, to start we need to know the three Pythagorean identities. 

1. Sin^2(x)+cos^2(x)=1 
2. Tan^2(x)+1= Sec^2(x)
3. 1+ Cot^2(X)=Csc^2 (x)

In these pythagorean identities as you see above,we can use number one to show why pythagorean identities is an appropriate name. 

In the Pythagorean theorem we use : A^2+ B^2= C^2
in the pythagorean identity we just simply substitute the sin and cos value in the A and B value.

For the more visual people here you go :
Pythagorean THEOREM  :    A^2   +  B^2         = C^2
Pythagorean IDENTITY  : Sin^2(x) + Cos^2(x) = 1

This makes sense because when we did the unit circle we learned that the radius otherwise known as C^2 will always equal 1. 

This is also shown in my diagram below . 

From this circle we also learned that sin and cos are the "legs" of out triangle. 

For example :

Because no matter what you do the bases, will always be bases and the radius will always be one the pythagorean identity is an appropriate name because connected to the pythagorean theorem. 

Aim : How do we simplify complex fractions ?

Argh.. ugly math again :(
but lets get started :

lets say one of your teachers gave you this problem :



So when dealing with complex fractions we have to solve it with parts.

First Step : Re-state your problem into parts.

4+ 1
     x
   
Second Step : In order to add fractions we need a common denominator.

For Example:

4+ 1
1    x  =   the common denominator would be x.

When you find the common denominator,you have to multiply the common denominator to the fraction that does not have it.

For example:
4 * x     = 4x
1 * x     =  x

Then, combine them
like this :
4x+ 1
  x

Third Step : After  you have done one part of the expression you go on to the second part.
3+ 2
     x2

Do the same steps as before, find a common denominator and multiply it to the fraction that does not have it.

So...  x2 is the common denominator.
So....

3   * x2  = 3x2
1   * x2  = x2

Unit them :

3x2 +2
 x2

Fourth Step : combine the whole expression.


4x+ 1  (÷)   3x2 +2
  x                x2

Fifth Step :  If a Expression has a diving symbol always use KFC, to change it to a multiplication.

K : keep
F : flip
C : change the bottom to the top

 K          F        C


4x+ 1     (÷)     3x2 +2
  x                       x2


Therefore :




4x+ 1   *      x2
  x             3x2+2

Sixth Step: Foil ALL possible pieces.

4x+1 = 4x+1
x       =   x



    X2      = (x) (x)
3x2+2    =  3x2+2

so :

4x+ 1   *  (x)(x)
   x           3x2+2

Seventh Step : Cross out what is repeated.

4x+ 1   *  (x)(x)
   x           3x2+2


P.s multiply x to 4x+1
Eight Step : Combine it all together for a final answer.

4x2+x

3x2+2

 A Problem :

 Works Cited:
Example Problem:  http://www.purplemath.com/modules/compfrac.htm
A Problem : http://www.regentsprep.org/Regents/math/algtrig/ATV2/simpcomplex.htm