Is Cos(x) = Cos(-x)? Exploring the Truth Behind Trig Functions

[phintastic]

all i need to know is whether or not the following is true:

-Cos(x) = Cos(-x)

i know that Cos(-x) = Cos(x), but i was just wondering if it was the same as -Cos(x). if anyone could help it would be greatly appreciated at this late hour ;)

 

[mjsd]

since cos(x)=cos(-x)

-cos(x) = cos(-x) can be true if cos(x) =0.

 

[HallsofIvy]

i know that Cos(-x) = Cos(x), but i was just wondering if it was the same as -Cos(x).

In other words, you are wondering if "A" is the same as "-A"? How much thought did you spend on this?!

Did you consider checking it on a calculator? Is cos(-10)= -cos(10)?

 

[phintastic]

ok thank you for your help. i forgot my calculator at school, and these trig functions can be tricky devils...

 

[dontdisturbmycircles]

Have you studied the unit circle? http://members.aol.com/williamgunther/math/ref/unitcircle.gif

For geometric reasons the y-coords are sin(x) and the x-coords are cos(x) since the radius of the circle is 1 for sin you can draw another side to the triangle formed by an angle and Sin(x) of course means opposite over hypotenuse so you have the height of the triangle (y coordinate) over 1, so its just the y coordinate. Similar reasoning shows that the x-coords are cos(x)

The neat thing about it is that you just have to memorize the 3 possible values for sinx and cosx, namely $$\frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}$$. And by picturing in your head where the tip of the angle would lie on the unit circle you can easily derive the values of most common values for all of the trig ratios!

Another one that helps is that tan(x) is the point where the tip of the angle eventually touches the line x=1.. So it becomes apparent that tan(x) is getting larger as x approaches $$\frac{\pi}{2}$$ without bound etc.

It would also easily answer your question since if the x coords are cos(x) its obvious that cos(-x) does NOT equal -cos(x)! It just equals cos(x) (unless x=0 but then you could come up with identities like $$5cosx=-3cos(-x) (x=0)$$ and what's the point of that.