Maths Trig. questions!!! help! (1 Viewer)

[pink_rabbit_<3]

Hey everyone!! im not sure if these are maths extension questions or 2 unit advanced questions coz i do both so i get confused. here are the questions..hope u can help!!

1a) simplify sin2X / 1 + cos 2X

b) hence, find the exact value of tan15

2. Find the exact value of tan 22.5 by using the expression for tan 2X

3. Prove that cos 3x = 4 cos³ x - 3 cos x

4. Find an expression for sin 3x in terms of sin x

THANKS =)

 

[vizman]

1. sin2x/ 1 + cos2x =
2cosxsinx/1+(2cos^2x - 1) =
sinx/cosx = tanx

 

[Riviet]

1 b) let x=15 (degrees)
Substitute into the result from part a):
tanx=sin2x / (1+cos2x)
tan15=sin30 / (1+cos30)
=(1/2) / (1+(rt3)/2)
=(1/2) / (2+rt3)/2
=1/(2+rt3)
=2-rt3 by rationalising the denominator.

2. tan2x=2tanx/(1-tan²x)
.'. tan45=2tan22.5/(1-tan²22.5)
tan²22.5 + 2tan22.5 - 1 = 0 since tan45=1
tan22.5=(-2+rt8)/2 by the quadratic formula
=-1+rt2

3. LHS=cos3x
=cos(2x+x)
=cos2x.cosx - sin2x.sinx
=(2cos²x-1).cosx - (2sinxcosx).sinx
=2cos³x-cosx-2sin²x.cosx
=2cos³x-cosx-2cosx.(1-cos²x)
=2cos³x-cosx-2cosx+2cos³x
=4cos³x-3cosx
=RHS ~ QED

4. Use a similar method as in 3. starting with sin(2x+x).

Hope that helps.

edit: sorry about that, vizman.

 

[vizman]

oh thanks riviet, just as i finished :mad1:

 

[pink_rabbit_<3]

thanx riviet!!!! and visman =) thats helps HEAPS!!!