why is cos(-4x)=cos(4x)? teach me concepts. can add on to sin(-4x) tan(-4x)

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- Aug 23rd 2010, 07:41 AM #1

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- Aug 23rd 2010, 07:52 AM #2
Its because cos(x) is an

**even**function (symmetric about the y axis), meaning that $\displaystyle f(-x)=f(x)$. Now, sin(x) is an**odd**function (symmetric about the origin), meaning $\displaystyle f(-x)=-f(x)$. You can see why this is the case by looking at the graphs of these two functions.

Now, tan(x) is an odd function, since $\displaystyle \tan (-x)=\dfrac{\sin (-x)}{\cos(-x)}=\dfrac{-\sin x}{\cos x}=-\tan x$.

Does this clarify things?

- Aug 23rd 2010, 09:04 AM #3

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Hello, stupidguy!

$\displaystyle \text{Why is }\,\cos(-4x)=\cos(4x)\;?$

We know: .$\displaystyle \cos\theta \:=\:\frac{adj}{hyp}$

Look at the graphs:

Code:| | * | hyp * | | * | | * 4x | - * - - - - - * - - | adj

Code:| | adj - * - - - - - * - - | * -4x | | * | | hyp * | | * |

Get it?

- Aug 23rd 2010, 09:16 AM #4

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- Aug 23rd 2010, 10:24 AM #5

- Aug 23rd 2010, 11:28 AM #6

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- Aug 23rd 2010, 12:24 PM #7

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One way to see it would be to find the Taylor series expansion of $\displaystyle f(x) = \cos{x}$ about $\displaystyle x = 0$ and note that all the powers of $\displaystyle x$ are even, but I realise that this is in the pre-Calculus section.

And this is why people should sign-up with appropriate names.

- Aug 23rd 2010, 12:33 PM #8

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If $\displaystyle 4x=135^0$, then $\displaystyle -4x=360^o-4x=360^o-135^o=225^o$

$\displaystyle 135^o=180^o-45^o$

$\displaystyle 225^o=180^o+45^o$

hence, both angles have the same horizontal co-ordinate on the unit-circle.

Cos(angle)=horizontal co-ordinate of a point on the unit-circle circumference,

hence it can be located on the horizontal axis.

If $\displaystyle 4x=220^o\Rightarrow\ -4x=360^o-220^o=140^o$

$\displaystyle 220^o=180^o+40^o$

$\displaystyle 140^o=180^o-40^o$

so the angles have the same horizontal co-ordinate.

Think of Cos(angle) as the horizontal co-ordinate of a point on the circle.

Sin(angle) is the vertical co-ordinate.

Tan(angle) is the slope of the line going through the origin (centre of unit-circle) and the point on the circle.

Or Tan(angle) is Sin(angle) divided by Cos(angle).

A positive angle is an anticlockwise movement starting at zero.

A negative angle is a clockwise movement starting at 360 degrees.