SOLUTION: 1/2cosx cannot equal cos1/2X Discuss how the 1/2 part affects the graphs of y=1/2cosx and y=cos1/2x I dont understand what each 1/2 does.

Click here to see ALL problems on Trigonometry-basics

Question 427310: 1/2cosx cannot equal cos1/2X
Discuss how the 1/2 part affects the graphs of y=1/2cosx and y=cos1/2x
I dont understand what each 1/2 does.
Found 2 solutions by stanbon, Theo:
Answer by stanbon(75887) (Show Source):
You can put this solution on YOUR website!
Discuss how the 1/2 part affects the graphs of y=1/2cosx and y=cos1/2x
----
In the form y = acos(bx) the a determines the amplitude;
the b determines the period.
---
Your Problem:
y = (1/2)cos(x)
The amplitude of the cos is normally "1".
Because of the !/2 the amplitude will be "1/2".
-----
y = cos((1/2)x)
The period of the cos is normally 2pi
Because of the 1/2 the period = (2pi)/(1/2) = 4pi
====================================================
Cheers,
Stan H.
==========

Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
(1/2) * cosine (x) is not the same as cosine ((1/2) * x).

1/2 * cosine (x) means you calculate the cosine of x and then you divide it by 2.

cosine ((1/2) * x) means you are dividing the angle by 2 and then taking the cosine of that.

assuming your angle is 60 degrees, then:

cosine (60) = .5

(1/2) * cosine (60) = (1/2) * .5 = .25

cosine ((1/2) * 60) = cosine (30) = .866025404

you can see the difference by looking at the triangles formed on the unit circle.

the picture of the unit circle is shown below:

in the unit circle, the 60 degree angle forms triangle ADE.

the cosine of 60 degrees would be equal to adjacent / hypotenuse which is equal to AE / AD.

since the hypotenuse of the triangles formed in the unit circle are always equal to 1, this means that the cosine of 60 degrees is equal to AE / AD which is equal to AE / 1 which is equal to AE.

half the cosine of 60 degrees would be half the length of AE.

when you half the angle of 60 degrees, you get the angle of 30 degrees.

the angle of 30 degrees on the unit circle forms triangle ABC.

the cosine of 30 degrees is equal to adjacent / hypotenuse which is equal to AC / AB.

since the hypotenuse of the triangles formed in the unit circle are always equal to 1, then cosine of 30 degrees equals AC / AB which is equal to AC / 1 which is equal to AC.

you have 1/2 the cosine of 60 degrees equal to 1/2 the length of AE and you have the cosine of 1/2 of 60 degrees equal to the cosine of 30 degrees equal to the length of AC.

that's the difference between 1/2 the cosine of an angle and the cosine of 1/2 of the angle.

hopefully that makes sense to you.

if not, let me know where you are still confused and i'll try to clear it up.