here's the question: solve for cos(2theta)=1
if you graph r= cos(2Theta)
then graph r=1
you find that there are four places where the graphs intersect.
however, when solved algebraically, there are two solutions which can be represented in an infinitely many format as: Theta = pi(K) (where k= an integer.)
so why do the graphs imply the answer should be Theta = pi(k)/2 ? What am I missing?
cos(2Ø) = 1
2Ø = 0 or 2Ø = 2π
Ø = 0 or Ø = π
Now here is the problem:
Are you graphing on a rectangular graph where Ø is the horizontal axis and r is the vertical axis
OR
Are you graphing in polar coordinates ???
I will illustrate:
let y = cos(2x), and y = 1
http://www.wolframalpha.com/input/?i=plot+y+%3D+cos%282x%29%2C+y+%3D+1+for+0+%3C+x+%3C+2%CF%80
notice the multiples of π intersections ---> kπ
now in polar:
http://www.wolframalpha.com/input/?i=plot+r+%3D+cos%282%C3%98%29%2C+r+%3D+1+
Don't know why the graphs are separate, let's blame Wolfram, but you can see they would intersect in 4 places
Does that shed some light on the situation?
To solve the equation cos(2theta) = 1 algebraically, we can use the double-angle identity for cosine, which states that cos(2theta) = 1 - 2sin^2(theta).
Setting this equal to 1, we have:
1 - 2sin^2(theta) = 1
Simplifying the equation:
-2sin^2(theta) = 0
Dividing both sides by -2:
sin^2(theta) = 0
Taking the square root of both sides:
sin(theta) = 0
Now, let's think about the graph of r = cos(2theta). Recall that the polar coordinate system represents points in terms of the distance from the origin (r) and the angle from the positive x-axis (theta).
When we graph r = cos(2theta), we are plotting points where the distance from the origin is equal to the cosine of twice the angle. Since cosine outputs values between -1 and 1, the graph of r = cos(2theta) only represents the points where -1 ≤ cos(2theta) ≤ 1.
On the other hand, when we graph r = 1, we are plotting points where the distance from the origin is always 1.
Now, let's analyze the intersection points. If cos(2theta) = 1, then r = cos(2theta) would also be equal to 1. This means that the points on the graph of r = cos(2theta) that intersect with the graph of r = 1 are the ones where r = 1.
So, the graphs intersect at the points where r = 1, which are infinitely many since the angle theta can have an infinite number of values that satisfy this condition.
When solving algebraically, we find that the solutions are given by theta = pi(k) where k is an integer. This is because each value of k represents a different angle that satisfies the equation sin(theta) = 0.
It's important to note that the two representations for the solutions, theta = pi(k) and theta = pi(k)/2, are both valid. They simply represent different sets of angles that satisfy the original equation cos(2theta) = 1. The graph r = cos(2theta) = 1 only displays the points where r = 1, hence the intersection points at theta = pi(k)/2.