Trig simplifying (1 Viewer)

[skillz]

CAn someone please help me with this.

Simply the following
sin^4x - cos ^4x



I'm really bad at doing these kinda simplyfying/ proof questions. does anyone have any tips on this kinda stuff?

 

[Riviet]

cos⁴x-sin⁴x=(cos²x)²-(sin²x)²
=(cos²x+sin²x)(cos²x-sin²x)
=cos2x

Know your identities well so you can apply them to manipuate and simplify trig expressions.

 

[pLuvia]

sin⁴x-cos⁴x
sin²x.sin²x-cos⁴x
(1-cos²x)²-cos⁴x
1-2cos²x+cos⁴x-cos⁴x
1-2cos²x
cos2x

 

[skillz]

another question thanks.

(secX-cosX)(cosecX-sinX)


Does secX.cosecx= cosec^2x?

 

[Riviet]

(secx-cosx)(cosecx-sinx)=[(1-cos²x)/cosx][(1-sin²x)/sinx]=
sin²x.cos²x
cosx.sinx

=sinx.cosx.
=1/2.(2sinx.cosx)
=1/2.(sin2x)

Does secX.cosecx= cosec^2x?

Nope, it equals 1/(sinx.cosx)

 

[skillz]

New question..

expand cos(x+2x) AND hence express cos 3x in terms of cos x.

i don't really understand the algebra in trig...

thanks

 

[Riviet]

cos(x+2x)=cosx.cos2x-sinx.sin2x
=cosx.(2cos²x-1)-2sin²x.cosx
=2cos³x-cosx-2cosx.(1-cos²x)
=2cos³x-cosx-2cosx+2cos³x
=4cos³x-3cosx

 

[skillz]

okay..

would, sinx + sinx = 2sin x?
sin2x+sin2x= 2sin2x OR sin4x?

 

[Riviet]

would, sinx + sinx = 2sin x?

Yes.

sin2x+sin2x= 2sin2x OR sin4x?

2sin2x. Remember that you can't just add the x's inside a trig expression.

 

[skillz]

Use the double angle formlua for tan 2x, and the fact that tan pi/4 = 1 to find the exact value of tan pi/8

Alright, i don't understand the difference between compound and double angle formula. someone please care to explain in words which i can understand?


thanks!

 

[pLuvia]

Compound angle is

cos(a+b)=cosacosb-sinasinb
cos(a-b)=cosacosb+sinasinb
sin(a+b)=sinacosb+cosasinb
sin(a-b)=sinacosb-cosasinb
tan(a+b)=[tana+tanb]/[1-tanatanb]
tan(a-b)=[tana-tanb]/[1+tanatanb]

Double angle is

cos2a
=cos²a-sin²a
=2cos²a-1
=1-2sin²a

sin2a=2sinacosa

tan2a=[2tana]/[1-tan²a]

Just read you wanted it in words so, compound angles are the formulae that requires you to have two angles, double angle it just has one angle. It kind of hard to explain maths in english if you know what I mean

 

[skillz]

Ic. do you have any tips on memorising these formulas?!

 

[rama_v]

Ic. do you have any tips on memorising these formulas?!

Just learn the first six on the list that pluvia has given you and then learn to derive the rest of the results. Its not too hard if you keep doing trig questions

 

[Riviet]

Do ALOT of questions with expanding and factorising these results, you will know them very well by the trials with constant practice.

A sexist way of memorising it:
Sine is straight, since the expansions of the compound angles of sin would be sc+cs, as the s's (sine's) like to be with the cos's since they're straight.
Cos is gay, since the expansions of the compound angles of cos would be cc+ss, as the c's like to be together [forcing the s's to go together]

Another way of remembering the sine expansions:
school cerficate + counter strike

 

[pLuvia]

For the double angles, just remember it as:
cos2x = cos(x+x), then you just use the cos(a+b) identity to solve it. But that's when you really forgot the identity

 

[Travis28]

Use the double angle formlua for tan 2x, and the fact that tan pi/4 = 1 to find the exact value of tan pi/8

Alright, i don't understand the difference between compound and double angle formula. someone please care to explain in words which i can understand?


thanks!

Can somebody solve this? I didn't want to make a new thread about it so sorry for the late bump.

 

[random-1005]

Can somebody solve this? I didn't want to make a new thread about it so sorry for the late bump.

tan2x= (2tanx)/(1-tanx^2)
therefore
tan45=(2tan22.5)/(1-tan22.5^2) but tan 45=1
therefore 2tan22.5=1-tan22.5^2 divide by two
tan22.5=(1-tan22.5^2)/2

i dnt think that simpfies further

 

[lyounamu]

tan2x= (2tanx)/(1-tanx^2)
therefore
tan45=(2tan22.5)/(1-tan22.5^2) but tan 45=1
therefore 2tan22.5=1-tan22.5^2 divide by two
tan22.5=(1-tan22.5^2)/2

i dnt think that simpfies further

Yes it can be simplifed further.

2tan22.5 = 1-tan22.5^2
tan^22.5 + 2tan22.5 - 1 = 0
treat tan22.5 = x
so x^2 + 2x - 1= 0
so using quad. formula:

x = (-2+sqrt(8))/2 = 0.4142125624.... (the other answer is rejected)
so tan22.5 = 0.4142...

 

[random-1005]

yes it can be simplifed further.

2tan22.5 = 1-tan22.5^2
tan^22.5 + 2tan22.5 - 1 = 0
treat tan22.5 = x
so x^2 + 2x - 1= 0
so using quad. Formula:

X = (-2+sqrt(8))/2 = 0.4142125624.... (the other answer is rejected)
so tan22.5 = 0.4142...

ignore this

 

[lyounamu]

ignore this

Read the question again. It's after an exact answer.