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Interpreting Expressions Involving Cosine Squared

Date: 03/16/2007 at 04:47:55
From: Mat
Subject: What does cos^2 actually mean?

I don't understand the difference between cos(x)^2, cos^2(x) and (cos
(x))^2.  Are they all same? 

I think I know what (cos(x))^2 is, and I think cos(x)^2 is not a 
valid term.  It should probably be cos(x^2).  So that leaves cos^2(x). 
I do not understand this term.  Could you please explain it?

Date: 03/16/2007 at 05:46:01
From: Doctor Peterson
Subject: Re: What does cos^2 actually mean?

Hi, Mat.

Cos^2 is an old traditional notation that we still retain, though it 
is inconsistent with modern functional notation in some respects.  It 
means the square of the cosine:

  cos^2(x) = (cos(x))^2

The expression is evaluated by finding the cosine of x and the
squaring the result.

To first square x and then find the cosine of the result, I don't
think it's quite illegal to write cos(x)^2, but it would be 
sufficiently misleading that we avoid it.  In particular, when you 
don't use the parentheses, cos x^2 means cos(x^2), not (cos x)^2. 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Definitions
High School Trigonometry

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