You are Here: Home >< Maths

# Sin(3x),(4x),(5x) or Cos(3x),(4x),(5x)??? watch

1. Can someone please explain to me how to use the double angle formulae to work these out?
2. (Original post by futureaussiecto)
Can someone please explain to me how to use the double angle formulae to work these out?
*gets out formulae booklet*

ok, so the single angle formula is;

sin (A + B) = sin A cos B + sin B cos A

which is in the formula book.

now ok, the double angle formula [sin(2x)] is basically;

sin (x + x) = sin x cos x + sin x cos x

which is also

sin (2x) = 2sin x cos x

so, if we were to look at

sin(3x)

this is basically;

sin(1.5x + 1.5x) = ...

so do the subbing in into the single angle formula;

2 sin (1.5x) cos (1.5x)

so therefore;

Sin(3x) = 2 sin 1.5x cos 1.5x

checking values;

let x=2

sin(3*2) = 2(sin (1.5*2) * cos (1.5*2)

which is right
3. see i understand that

but in my book it says

Sin3A

=sin(2A+A)
=Sin2AcosA+cos2ASinA

which threw me
4. No you have to find it in terms of sinx.
Basically, use the double angle formulae to express sin3x in terms of sin2x and sinx, then use the double angle formulae again to get the sin2x down to sinx.
5. (Original post by futureaussiecto)
see i understand that

but in my book it says

Sin3A

=sin(2A+A)
=Sin2AcosA+cos2ASinA

which threw me
Yes that is true.
6. That formula from your book is just using the addition formulae instead of the double angle ones, perfectly acceptable, and a bit easier for odd numbers I guess. A and B are just numbers, if you get 5A just let it be C or something and carry on. Nothing sacred about letters.
7. (Original post by futureaussiecto)
Can someone please explain to me how to use the double angle formulae to work these out?
sin(2x)=sin(x+x)=sinxcosx+cosxsi nx=2sinxcosx
cos(2x)=cos(x+x)=cosxcosx-sinxsinx=cos^2x-sin^2x
sin(3x)=sin(2x+x)=sin2xcosx+cos2 xsinx,,,now use expressions for cos(2x) and sin(2x)
cos(3x)=cos(2x+x) etc
sin(4x)=sin(2x+2x) or sin(3x+x) etc
8. (Original post by futureaussiecto)
see i understand that

but in my book it says

Sin3A

=sin(2A+A)
=Sin2AcosA+cos2ASinA

which threw me
ahh

ok, i did this;

sin (1.5x + 1.5x) = ...

all they did was say

sin (2x + x) = ...

and worked it out like that instead.

in both cases, the 3x was split up into two parts, but they were split differently.
9. The idea is you break it down so there's only terms involving sinx or cosx, no sin2x or sin3x etc. So just use the addition formulae to get it in terms of sin2x and cos2x, and the double-angle formulae to break it down further. It just involves a lot of substitution.
10. so sin(1x) is just sin x?

if so, isnt sin(3x) just sin(2x) [which is 2sinxcosx} + sinx??
11. (Original post by futureaussiecto)
so sin(1x) is just sin x?
Yes/
if so, isnt sin(3x) just sin(2x) [which is 2sinxcosx} + sinx??
No/
12. (Original post by futureaussiecto)
if so, isnt sin(3x) just sin(2x) [which is 2sinxcosx} + sinx??
No:

sin(3x)
= sin (2x + x)
= sin2x.cosx + cos2x.sinx
= 2sinx.cosx.cosx + (1-2sin2x)sinx
= 2sinx.cos2x + sinx - 2sin3x
= 2sinx(1-sin2x) + sinx - 2sin3x
= 2sinx - 2sin3x + sinx - 2sin3x
= 3sinx - 4sin3x
13. (Original post by Libertine)
No:

sin(3x)
= sin (2x + x)
= sin2x.cosx + cos2x.sinx
ok from that it looks like

sin2x = sin2xcosx
and
sinx = cos2xsinx

14. (Original post by futureaussiecto)
ok from that it looks like

sin2x = sin2xcosx
and
sinx = cos2xsinx

sin (2x + x) does not expand to sin2x + sinx

Never do that with trig functions.
15. (Original post by Dekota)
sin (2x + x) does not expand to sin2x + sinx

Never do that with trig functions.
yeh i know, i was just saying from the prev quote that when it said

sin(2x + x) = sin2x.cosx + cos2x.sinx

that (2x) is the sin2x.cosx bit

and

(x) is the cos2x.sinx
16. (Original post by futureaussiecto)
yeh i know, i was just saying from the prev quote that when it said

sin(2x + x) = sin2x.cosx + cos2x.sinx

that (2x) is the sin2x.cosx bit

and

(x) is the cos2x.sinx
But it's not. The whole expression on the right is equal to the whole expression on the right. The bits before and after the addition signs don't correspond.

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: February 1, 2006
Today on TSR

### The TSR A-level options discussion thread

Choosing A-levels is hard... we're here to help

### University open days

• Heriot-Watt University
School of Textiles and Design Undergraduate
Fri, 16 Nov '18
• University of Roehampton
Sat, 17 Nov '18
• Edge Hill University
Faculty of Health and Social Care Undergraduate
Sat, 17 Nov '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams