Deriving a Formula for cos 3a Using Only Cosine - Step by Step Guide

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Homework Help Overview

The original poster attempts to derive a formula for cos 3a using only cosine functions. The problem involves trigonometric identities and requires manipulation of cosine and sine functions to express cos 3a solely in terms of cosine.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the application of the cosine addition formula and explore the steps taken to derive the expression for cos 3a. There are questions about the correctness of signs and the application of trigonometric identities. Some participants express confusion about the derivation process and seek clarification on specific steps.

Discussion Status

The discussion is ongoing, with participants providing corrections and suggestions for the original poster's approach. There is recognition of a typo in the application of the cosine addition formula, and further exploration of the derivation is encouraged. Multiple interpretations of the steps are being examined, but no consensus has been reached.

Contextual Notes

Participants are working under the constraint of deriving the formula using only cosine, which influences their approach and reasoning. There is also a focus on correcting earlier mistakes without resolving the overall derivation challenge.

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Homework Statement


Derive a formula for cos 3a which would only involve cosine on the right side.


Homework Equations


cos2x = cos(x)^2 - sin(x)^2
cos2x = 2cos(x)^2 - 1
cos2x = 1-2sin(x)^2


The Attempt at a Solution



cos(3a) = cos(2a + a)
= cos2acosa + sin2acosa
= (2cos(a)^2 - 1)cosa - (2sinacosa)cosa
= 2cos(a)^3 - cosa - 2sina cos(a)^2


Am I on the right path? Or should I just continue to do keep doing this?
 
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cox(x+y)=cosxcosy-sinxsiny ...so your sign is wrong
 
Sorry. My bad, It was such an obvious mistake.

Here it goes:

cos(3x) = cos(2x + x)

= cos2xcosx - sin2xsinx
= (2cos(x)^2 -1)cos(x) - (2sin(x)cos(x))cos(x)
= 2cos(x)^3 - cos(x) - 2sin(x)cos(x)^2
...

Ok. Now I am confused again. Please help me. Thanks.
 
That sign error was clearly a typo since you corrected it in the next line.

Your real problem is not the sign:
cos(3a) = cos(2a + a)
= cos2acosa + sin2acosa
No, that is cos(2a)cos(a)- sin(2a)sin(a)

= (2cos(a)^2 - 1)cosa - (2sinacosa)cosa
= 2cos(a)^3 - cosa - 2sina cos(a)^2

should be = (2cos^2(a)- 1)cos(a)- (2sin(a)cos(a))sin(a)
= 2cos^3(a)- cos(a)- 2sin^2(a)cos(a)

Now replace sin^2(a) with 1- cos^2(a).
 

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