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