ok I was doing this problem
cos^-1 (0)
and was like oh that would be pi/2 but then I thought about it some and decided that even though it's right it can't possibly be...
if tan(pi/2) = the first derrivative of a vertical line = undefined
well not really there's a deffintion correct it has a value... a number that approaches zero (from both directions?)... wow uh?... but is not zero becasue the line exists and in order for it to be in existence... there is a first derrivative... right so it's not that it doesn't exist... just not defined... there's a value... that I could find some way to define would... uh... hm....
but ya anways so I thoguht about that and was like so then why on earth do we define cosine of a vertical line (pi/2) to be zero... it's not zero because it's in existence just incredably small... aproaches zero from both sides... interesting... just like tangent shouldn't then have some true value that's not zero that we should leave undefined...
I know i know... by defintion it's zero put in your calculator... ya but if you think about it it's not really zero is it it's really undefined...
unless those are two different versions of undefined??
the width of a vertical line that has no width... but exists so therefore has one just like the first derrivative of it...
well I'm confused... can you please help me understand as just thinking about it leads me to believe that cosine pi/2 is really undefined just like tangent pi/2 is... why is it not undefined shouldn't be... why do we say it's zero casue it's really not now is it...???
cos^-1(0) is the angle(s)for which the cosine is zero.
pi/2 (radians) is one such angle.
So is 3 pi/2. There are others at multiples of pi.
There are no real values of x for which cos x is undefined.
I understand your confusion, and I'm here to help clarify it for you. Let's break it down step by step.
First, let's remember the definition of the inverse cosine function, often denoted as cos^(-1) or arccos. The inverse cosine of a number x returns the angle whose cosine is x. In other words, cos^(-1)(x) = θ if and only if cos(θ) = x.
Now, let's focus on your specific question: cos^(-1)(0). According to the definition, we need to find the angle whose cosine is equal to 0. When you think about it, the cosine function represents the x-coordinate of a point on the unit circle. So, where does the x-coordinate equal 0 on the unit circle? It happens exactly at π/2 and 3π/2, which correspond to the points (0, 1) and (0, -1) respectively.
Therefore, cos^(-1)(0) can be either π/2 or 3π/2. Both angles have the same cosine value of 0.
Now, let's address your observation about the tangent function and its derivative. The tangent function, tan(x), is defined as the ratio of the sine and cosine functions: tan(x) = sin(x) / cos(x).
You correctly stated that the derivative of a vertical line is undefined. In calculus, the derivative of a function measures its rate of change at a given point. For a vertical line, the slope (or rate of change) is infinite, which means it is undefined.
However, this does not necessarily imply that the tangent function itself is undefined at π/2. The tangent function is defined for all real numbers except the values where cosine is equal to zero. This is because dividing by zero is undefined in mathematics.
While the derivative of the tangent function is undefined at π/2, it does not mean that the tangent function itself is undefined at that point. Instead, it is a location where the function approaches infinity or negative infinity. Hence, we say that the tangent of π/2 is undefined because its denominator, cosine, equals zero, but we also know that the limit as x approaches π/2 from both sides is positive or negative infinity.
Finally, let's examine the value of cosine at π/2. By definition, cos(π/2) is equal to zero. It signifies that the x-coordinate of the point (0, 1) on the unit circle is zero. Despite it being small or approaching zero, we still define it as zero because that is the convention established in trigonometry.
To summarize, the inverse cosine of 0, cos^(-1)(0), can have multiple solutions, which are π/2 and 3π/2. The tangent of π/2 is undefined because its denominator, cosine, becomes zero, but it does not mean that the function itself is undefined. Lastly, cosine is defined as zero at π/2 because that is the convention adopted in trigonometry.