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What is cos infinity?

Updated: 9/24/2023
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Q: What is cos infinity?
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The range for y = 4 cos (2x) is [-4, +4].Not asked, but answered for completeness sake, the domain is [-infinity, +infinity].


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If I understand your question correctly, such a sequence is an = x cos(πx). It has neither an upper nor lower bound. It's divergent, but its limit is neither infinity nor negative infinity.


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The answer depends on the side from which x approaches 0. If from the negative side, then the limit is negative infinity whereas if from the positive side, it is positive infinity.


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Note: Assuming you are working with natural, integer, rational(fraction), or real numbers. It doesn't. Infinity is not a number, even though, due to us mathematicians being lazy, we denote something = infinity. But we NEVER write tan 90 = infinity. But rather lim_x->(90degree) tan x = infinity. Meaning as x gets closer to 90 degree (even though degree is a horrible measurement for angle, we will use it), the value of tan x gets large faster and unbounded. tan x? It doesn't exist. Why? Because tan x is defined as (sin x / cos x). When x = 90 degree, cos x = 0, while sin x is positive around x = 90 degree. sin x / cos x := sinx x 1 / cos x, x = 90 degree, we get 1 x 1 / 0. But the definition for inverses does not include 0, meaning 1 / 0 does NOT exist. so, sadly, tan 90degree doesn't exist. The best we can do is see what happens around x = 90degree for tan, as you go into Calculus, you will know the tool is called limits and derivatives. And you might also see the reason it is +infinity, but not -infinity. (tan x approaches -infinity as x approaches 180degree). WHat's more? You will learn a far better measurement for angle that you will stick with in Calculus.


Sin x cos x equals 0?

For the product to be zero, any of the factors must be zero, so you solve, separately, the two equations: sin x = 0 and: cos x = 0 Like many trigonometric equations, this will have an infinity of solutions, since sine and cosine are periodic functions.


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On the unit circle at 90 degrees the 90 degrees in radians is pi/2 and the coordinates for this are: (0,1). The tan function = sin/cos. In the coordinate system x is cos and y is sin. Therefore (0,1) ; cos=0, & sin=1 . Tan=sin/cos so tan of 90 degrees = 1/0. The answer of tan(90) = undefined. There can not be a 0 in the denominator, because you can't devide by something with no quantity. Something with no quantity is 0. Or, on a limits point of view, it would be infinity.