Turn on thread page Beta
    • Thread Starter
    Offline

    0
    ReputationRep:
    Hi!

    Im finding trigonometric identities relatively hard, and Im wondering if anyone can help me solve the question below:

    (X replaces THETA)


    Code:
    Show that 8Cos^4X = cos4X + 4cos2X + 3
    Thanks!
    Offline

    8
    ReputationRep:
    Hi!

    Im finding trigonometric identities relatively hard, and Im wondering if anyone can help me solve the question below:

    (X replaces THETA)
    Use this identity multiple times

     cos^2(\theta) = \frac{cos(2\theta)+1}{2}
    Offline

    3
    ReputationRep:
    (Original post by spread_logic_not_hate)
    Use this identity multiple times

     cos^2(\theta) = \frac{cos(2\theta)+1}{2}
    I can't fault logic's method, but you did specify FP2.

    Part of the FP2 idea is to be able to use De Moivre to expand cos(nx) or sin(nx). You then get some polynomial in cos and/or sin to play with.
    • Thread Starter
    Offline

    0
    ReputationRep:
    (Original post by ian.slater)
    I can't fault logic's method, but you did specify FP2.

    Part of the FP2 idea is to be able to use De Moivre to expand cos(nx) or sin(nx). You then get some polynomial in cos and/or sin to play with.
    Yeh!, I need to use De Moivre.

    Can you help?
    Thanks!
    Offline

    3
    ReputationRep:
    OK

    exp(ix) = cos(x) + isin(x) by definition

    so exp(nix) = (cos(x) + isin(x))^n

    but also exp(nix) = cos(nx) + isin(nx)

    If you expand say cos(4x) + isin(4x) = (cos(x) + isin(x))^4

    you equate real parts to get cos(4x) and imag parts to get sin(4x).

    Then the examiner dreams up ways to get you to use that
    Offline

    8
    ReputationRep:
    Yeh!, I need to use De Moivre.

    Can you help?
    Thanks!
    Ah sorry did not know that. Ok then, De Moivre's theorem is

     (cos x + i sin x)^n = cos(nx) + i sin (nx)

    In your case i'd write this

     (cos x + i sin x)^4 =  [(cos x + i sin x)^2]^2 = cos(4x) + i sin (4x)

    Now expand the LHS and collect all the real terms, which will be equal to cos(4x)...
    Offline

    3
    ReputationRep:
    In the question, I'd use DM to expand cos(4x) (remembering that sin^2 = 1-cos^2) and also cos(2x) (although you could just quote that). Work out the RHS and hope when the smoke clears you're left with the LHS
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: February 11, 2010
Poll
Do you think parents should charge rent?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.