Double Angle Formula (1 Viewer)

[hsc2013]

I just want to ask you guys a few questions.Firstly, the double angle formula for cos has 3 different types. How do people know which one to use? I also want to know how do you change a product to sum or differences. Lastly, do questions specifically state when to use the auxiallary angle method or do i just have to know when to use it. Could you guys provide me with some example questions and solutions especially on my question about the double angle formula, so i could understand.

Thanks

 

[812]

I just want to ask you guys a few questions.Firstly, the double angle formula for cos has 3 different types. How do people know which one to use?

Look for things that can cancel out.

If the question was simplify 1 + cos2x
then you would use the formula cos2x = 2cos^2(x) - 1 because + 1 and -1 would cancel out.
i.e. 1 + cos2x
= 1 + [2cos^2(x) - 1]
= 2cos^2(x)

Another situation would be: if everything else was in terms of sin, then you would use 1 - 2sin^2(x)
same with if everything was in terms of cos, then you would use 2cos^2(x) - 1

Hope that helps.

 

[IamBread]

It doesn't really matter which one you use if you know how to derive each of them.







So what I'm trying to say is, it doesn't matter which one you use. If you can remember these rules, you can keep using until you can get it into the form you can do something with.

 

[Kimyia]

Look for things that can cancel out.

If the question was simplify 1 + cos2x
then you would use the formula cos2x = 2cos^2(x) - 1 because + 1 and -1 would cancel out.
i.e. 1 + cos2x
= 1 + [2cos^2(x) - 1]
= 2cos^2(x)

Another situation would be: if everything else was in terms of sin, then you would use 1 - 2sin^2(x)
same with if everything was in terms of cos, then you would use 2cos^2(x) - 1

Hope that helps.

+1.This pretty much sums it up.

 

[hsc2013]

Thank you guys. I also need help with proving identities. Could you guys provide me with some examples from easy Q's to hard Q's. Als is it ok if i post a few hard questions that i could not do from a textbook?

 

[Carrotsticks]

Thank you guys. I also need help with proving identities. Could you guys provide me with some examples from easy Q's to hard Q's. Als is it ok if i post a few hard questions that i could not do from a textbook?

Yes it's okay.

 

[812]

Thank you guys. I also need help with proving identities. Could you guys provide me with some examples from easy Q's to hard Q's. Als is it ok if i post a few hard questions that i could not do from a textbook?

Sure, go ahead.

For proving identities, you just need to do a wide range of questions so you don't get stuck with something you've never seen before during the exam.

 

[hsc2013]

Ok, These questions that I am asking to you guys I dont know how to do them so please help me solve them so I can do these questions again if they come out in the exam.

54. tan(x+45)tan(x-45) =1

55. tan2x - tanx
___________ = tan^2x
tan2x + cotx

43. cos4x = 8cos^4x - 8cos^2x +1

40. tan(45+a) + tan(45 -a) = 2/cos2a

50. sin^3x +cos^3x
_____________ = 1- 1/2sin2x
sin x +cos x

51. tan(x + a)tan(x - a) = tan^2x - tan^2a
_____________
1 - tan^2xtan^2a

I also want to ask you guys what is the difference between these 2 equations: tan(x-45)tan(x+45) = 2 and tan(x+45) + tan(x -45 ) =2
I just want to know what is the solution and the working out . Please note that the RHS of this question I am asking about these 2 equations are wrong, I just want to know how to write the working out though.

Thank you

Note : question 55, 51 and 50 are fractions. Sorry about the inconsistency

 

[deswa1]

Sorry, do you want us to solve these or to prove them as an identity? The reason is that the first one isn't an identity (sub in x=30 for example). 43 though is. The way to do 43 (assuming you don't do 4U and can't use De Moivre's theorem) is to turn cos4x into cos(2x+2x), expand that to get cos^2(2x)-sin^2(2x). The you expand that into single angle expressions and then use sin^2(x)=1-cos^2(x) to get everything into powers of cos. Have a go now and see if you can do it

 

[hsc2013]

Sorry, The question states to prove as an identity

 

[deswa1]

The problem with that is that the first isn't an identity so by definition you can't prove it. Which book did you get that question from?

 

[hsc2013]

Sorry man, I dont know what do you mean by expand to single expressions?

 

[deswa1]

All right, give me a minute and I'll type it up...

Will update as I go

The formula I will be using is cos(2x)=2cos^2(x)-1:

<a href="http://www.codecogs.com/eqnedit.php?latex=cos4x=cos(2x@plus;2x)\\ =2cos^2(2x)-1 \\ =2(2cos^2(x)-1)^2-1 \\ =2(4cos^4(x)-4cos^2(x)@plus;1)-1 \\ =8cos^4(x)-8cos^2(x)@plus;1 \\ =RHS" target="_blank"><img src="http://latex.codecogs.com/gif.latex?cos4x=cos(2x+2x)\\ =2cos^2(2x)-1 \\ =2(2cos^2(x)-1)^2-1 \\ =2(4cos^4(x)-4cos^2(x)+1)-1 \\ =8cos^4(x)-8cos^2(x)+1 \\ =RHS" title="cos4x=cos(2x+2x)\\ =2cos^2(2x)-1 \\ =2(2cos^2(x)-1)^2-1 \\ =2(4cos^4(x)-4cos^2(x)+1)-1 \\ =8cos^4(x)-8cos^2(x)+1 \\ =RHS" /></a>

Does that make sense?

 

[hsc2013]

The book I got it from was Maths in focus 1. Anyways I still want to know how to split the LHS because I dont know how to solve an equation with a multiple in the question as stated in question 54. Although, could you please help me with the rest?

 

[hsc2013]

Thank you

 

[deswa1]

Assuming 54 actually was an identity, you would do it like any other one, just multiply them. It would look like this:

tan(x+45)tan(x-45)=(tanx + tan45)/(1-tanxtan45) TIMES (tanx - tan45)/(1+tanxtan45)
=(tanx+1)/(1-tanx) TIMES (tanx-1)/(1+tanx) [tan45=1]
=(tan^2(x)-1)/(1-tan^2(x))

 

[Darkraiider66]

"I just want to ask you guys a few questions"

yes officer, anything you say...

 

[Darkraiider66]

"How do people know which one to use?"

you'll find that this pops up a lot in previous question one integration questions, in this case just look at what you have, and what you can integrate useing the table of standard integrals.

 

[812]

Ok, These questions that I am asking to you guys I dont know how to do them so please help me solve them so I can do these questions again if they come out in the exam.

54. tan(x+45)tan(x-45) =1

55. tan2x - tanx
___________ = tan^2x
tan2x + cotx

43. cos4x = 8cos^4x - 8cos^2x +1

40. tan(45+a) + tan(45 -a) = 2/cos2a

50. sin^3x +cos^3x
_____________ = 1- 1/2sin2x
sin x +cos x

51. tan(x + a)tan(x - a) = tan^2x - tan^2a
_____________
1 - tan^2xtan^2a

I also want to ask you guys what is the difference between these 2 equations: tan(x-45)tan(x+45) = 2 and tan(x+45) + tan(x -45 ) =2
I just want to know what is the solution and the working out . Please note that the RHS of this question I am asking about these 2 equations are wrong, I just want to know how to write the working out though.

Thank you

Note : question 55, 51 and 50 are fractions. Sorry about the inconsistency

54. Use compound angle formula for tan on the LHS.
55. Use double angle formula for tan on the LHS where possible and cancel things out.
43. Use the double angle formula for cos twice, then expand the square.
40. Use compound angle formula for tan. Combine the two fractions. Expand and simplify the top. Factorise the top. Use the t formula for cos.
50. Use sum of two cubes formula for the numerator. Cancel. Use sin^2a + cos^2a = 1 identity to simplify. Then use reverse double angle formula for sin.
51. Compound angle formula for tan.

These questions aren't hard at all. Some of them require the use of other formulas (sum of two cubes) but those should be very familiar to you by now. The rest is just subbing in the appropriate formulas.