Unit 5- proving identities

Hi everyone. This is manveer Today I will be blogging about the things that we went over in class but very briefly. I don't like to talk to much so I will go straight into it.

Proving Identities :

Know these strategies :  
1) Use known identities i.e. ( sinx/cosx = tanx)
2) Use the Pythagorean Identity or one of its alternate forms
3) Rewrite the equation/expression using sin and cos
4)Multiply the numerator and denominator by the conjugate
5) FACTOR

So let's begin with the first question .

To determine the non-permissible values we need to know when sinx ≠ 0 .

x ≠ 0 ,180 , 360

The general solution for this equation would be x≠180n , where nEI

To verify the equation we need to substitute our x values. In this case x=30

1-sin²x  = sinxcosxcotx

L.H.S    = 1-sin²(30)

             = 1-(1/2)²

             = 1- 1/4

             = 3/4

R.H.S   = sin(30)cos(30)cot(30) 

            = (1/2) ( √3/2) (√3)

            = 3/4

So now according to the results we can tell that L.H.S = R.H.S

Lastly , to prove that the identity is true for all the permissible values of x we need to simply the equation.
1-sin²x  = sinxcosxcotx
cos²x = sinx * cosx * cosx/sinx
cos²x = cosx*cosx
cos²x = cos²x

That's it !

Since this was a tricky question and it was a TRAP for all of us, I thought of reminding you all again.

tan(π/2 - θ ) = cotθ

All we need to do is REPLACE θ with π/2 - θ

tan(π/2 - θ) = sin(π/2 - θ) / cos (π/2 - θ)

                   = cosθ / sinθ

                   = cotθ

There are a lot of steps for this question but I have 6 easy steps that could help you get the answer.

1) Break sin3x into sin2x + sinx

2) α=2x & β=x

3) Replace sin2x => 2sinxcosx

4) Replace cos2x => 1-2sin²x

5) Multiply inside the brackets

6) Combine like terms

MAKE SURE YOU ADD THE BRAKCETS !



here are the double angle identities don't forget them :) 

ok bye

Unit 5 - Trigonometric Identites

Hey everyone ! This is Jaspreet (:

First of all , 43 days left till the provincial exams ! I know Mr.P reminds us this everyday so I thought of reminding you again :)

Today this blog is going to be about Trigonometric Identities. So far everything is going alright but I know its going to get harder . Before I begin lemme tell you all that I am not a very good person when it comes to explaining things like this, unless I actually do it but I'll try my best! (:

Let's start off with something cute :)

Here is a brief review of the things that we did yesterday ! (Its a lot of formulas!)

Sum and Difference Identities

Don't worry the formulas on top are on your formula sheet ! (:

 

 

 

To write an expression as a single trigonometric function, you need to know "α" & "β" and the correct formula.

 

In this example ;

 α = 45°

 β = 17°

Therefore, sin48°cos17° - cos48°sin17°

                 = sin(α - β)

                 = sin (48°-17°)

                 = sin31°           

 

                           

Moving on to today's lesson. Today we talked about Double Angle Trigonometric Identities.

 

Double Angle Trigonometric Identities

 

Double angle identities:



 

 

      

 

Sine & Cosine of a Double Angle

 

       Since,  sin2α = sin (α + α)

                   sin2α = sin α cos α + cos α sin α
Therefore,  sin2α = 2sin α cos α

The cosine formula is just as easy:

       Since,  cos2α = cos (α + α)
                   cos2α = cos α cos β - sin α sin β
Therefore,  cos2α = cos²α -sin²α



There are two other formula's for cos2α

 

1.) cos2α = cos²α − sin²α

We can replace "cos²α" with " 1-sin²α" because according to the Pythagorean identity , cos²θ + sin²θ = 1.

 

Therefore ,  sin²θ = 1 - cos²θ

                               &

                   cos²θ = 1 - sin²θ

 

Moving back to cos2α = cos²α − sin²α

                          cos2α = 1 - sin²α - sin²α

                          cos2α = 1- 2sin²α

 

2.) cos2α = cos²α − sin²α

Instead of replacing "cos²α" , we are going to replace "sin²α".

            [ Remember to add the brackets!!!! ]

 

                  cos2α = cos²α − sin²α

                  cos2α = cos²α − (1 − cos²α)

                  cos2α = cos²α − 1 + cos²α

                  cos2α = 2cos²α − 1

 

 

 

To write an expression as a single trigonometric function using double-angle identities you need know your " α "

 

For this example ;

 

  2tan76°                         α = 76°

 1-tan²76°

 

  2tanα°             

 1-tan²α°   = tan2α

                 = tan 2 * 76°

                 = tan154°

 

 

This was probably the hardest part of this unit so far , which confused most of us at first but eventually we got it ! {THANKS TO MR.P } (: Although I'm not a 100% confident about this part but I will try to explain the best I can .

 

For this example ;

Part A ) We need to find the permissible values for the expression

Part B ) We need to simplify the expression to one of the 3 primary trigonometric functions { sinx , cosx , tanx }

 

I have clearly shown all the steps in the picture but I will give a brief explanation of what needs to be done.

 

So for the first part, we need to find all the values where sinx=0 

 sinx = 0 , π , 2π , .......

 x= πn , where n∈I

 

Then we need to find all the values where cosx=0

cosx = π/2 , 3π/2 , ........

x=π/2 + πn , where n∈I

 

To get general solution for this equation we need to combine the two solutions together.

 General solution : x= π/2 n , where n∈I

 

∴ All the permissible vales for the expression 1-cos2x  are the real numbers except

                                                                            sin2x

    when x= π/2 n , where n∈I

 

 

 

Last but not the least , determining exact trigonometric value for angles .

Note : When finding exact vales KEEP YOUR CALCULATORS AWAY !

 

 

 

Finding the exact value is probably the easiest thing so far.

 

To find the exact value for : 2sinπ/12 cosπ/12                    [ α=π/12 ]

= 2sinαcosα

= sin2α

= sin2 * π/12

= sin π/6

=1/2

 

Well that's all for today ! We have learnt a lot so far and there's more to go >.< (45-20 = 25 more pages to go ! ) I hope this blog helped everyone who is away and also those people who missed today's class .

Good luck and goodnight <3

 



Hi everyone ! It's Ella.

I'll be summarizing what we have learnt today. The topic was translation of sine and cosine functions. To start it off, let me recall about what Mr. Piatek discussed last week.

Transformations of some and cosine functions
The formulas are:
y = asinbx and y = acosbx

In vertical stretches :
• amplitude changes from the basic of 1 to |a|.

Amplitude is equal to maximum - minimum divided by 2.

•if a < 0, the function is reflected through the horizontal middle axis of function.

In horizontal starches:

• period changes from the basic of  2p to 

•if b < equal to 0, the function is reflected in the y - axis.

Now going to translation of sine and cosine functions.

y = asinb(x-c)+d and y = acosb(x-c)+d

Horizontal translations

•if it's negative, the function shifts c units to the right. 

•if it's positive, the function shifts c units to the left.

•it's called the phase shift.

vertical translations

• called vertical displacement. It is the result of change in the middle axis.

•if it's positive, the function shifts d units up.

•if it's negative, the function shifts d units down.

Order for applying transformations:

•perform all horizontal stretches and reflections.

•perform all vertical stretches and reflections.

•perform all translations.

Here's an example:

Now going to graphing the tangent functions

The graph of the tangent function, y = tanx, is periodic, but it is not sinusoidal. Periodic, meaning that it will repeat itself over regular intervals (cycles) of it's domain. Not sinusoidal, meaning that it does not fluctuate back and forth like a sine or cosine graph. 

Here's an example of tangent graph :

Hello guys.. This is Sargun and Todays blog will be about Trignometric Identites!!

What are trigonometric Identities??

In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. Geometrically, these are identities involving certain functions of one or more angles. ...

Reciprocal identities

Sum and difference of identities

These are the some values that you need to know..

Angle	Sin	Cos	Tan=Sin/Cos
30°			1/√3 = √3/3
45°			1
60°			√3

Transformations Golf

It has been a pleasure marking your Transformations Golf Assignment.
You should all be proud of your work.
Here are your creations:

Julan, Manpreet, Daniel, and Rhenz

Melissa, Andrew, Jasdeep, and Carl

Rowsell, Jade, Mydee, and Marielle

Jaspreet, Navdeep, Ella, ans Manveer

Ria, Rachel, Gabi, and Kenneth

 

 

Alesandro, Raphael, Albien, and Jerricson