Well (1/2)^2 = 1/4, so you should just be ruling a horizontal line at y = 1/4 and show that it crosses the graph y = cos(x) infinitely many times.
y = cosx, y = (1/2)^x
- State four points where cosx = (1/2)^2
- Also, why are there an infinite number of these points?
I've gotten the graphs but except for x = 0, I don't know which other points to choose. The graph can be seen below:
I need to get this in by tomorrow so if any of you wonderfolks can help me out with this, that'd be great. Thanks ofcourse!
On a sidenote: Would it be right to say that sinx ≤ x when x > 0?
That was a typo- he meant . Bilal7, since is always positive it will cross cos(x) an infinite number of times. You will have to use a numerical method to approximate solutions. You could, for example, use "Newton-Raphson": to solve take the derivative: and form the sequence , choosing a starting from your graph that is reasonably close to a root.
Let's start with the side note. $0 < x \implies sin(x) < x\ and\ 0 \le x \implies sin(x) \le x.$ Proof follows.
$Let\ z = x - sin(x).$
$sin(x) \le 1\ \implies - 1 \le - sin(x) \implies 0 < x - sin(x)\ if\ 1 < x \implies sin(x) < x\ if\ 1 < x.$
$sin(0) = 0 \implies sin(x) = x\ for\ x = 0.$
$Assume\ 0 < x \le 1.$
$So\ cos(x) < 1 \implies - cos(x) > - 1 \implies 1 - cos(x) > 0.$
$z = x - sin(x) \implies \dfrac{dz}{dx} = 1 - cos(x) > 0 \implies z\ is\ increasing \implies z > 0 \implies x - sin(x) > 0 \implies sin(x) < x.$
As for your main question, I have no idea what it is. What you say in your title and what you say in your text are different. And are you being asked for exact answers or approximate answers? Please state the problem completely and exactly.
Yes, there was a typo in the thread title and the second function is infact y = (1/2)^x, and the question doesn't state if they should be exact or approximate answers so I guess either would be fine.
@Halls of Ivy- We haven't covered that "Newton-Raphson" method so I went back and used a graphing tool and here are the results:
Point 1: (0, 1)
Point 2: (1.08, 0.47)
Point 3: (4.75, 0.04)
Point 4: (7.85, 0)
Look good? This is how they did it in the textbook too. Also, there are an infinite number of points because y=(1/2)^x is always positive, right?
Here is a better plot.
BUT, I really am curious about the point of this question?
What concept and/or principle is the author trying to make?
Surely it is nothing to do with the period of $\cos(x)~???$
Well this problem seems to have little point unless it is to get you to find approximate answers using a graphing calculator.
Trigonometry comes up all over advanced math, and it comes up in more forms of practical life than just architecture. (I do not have any vested interest in trig; I was never a math teacher or a mathematician.) The point is that a lot of high school math is giving you tools for things that come later, and it is really hard to know what tools you will need later in life. I studied history in college, but found myself using math in terms of my career.