Can you explain what the sin of infinity, cos of infinity, and tan of infinity is, and why?

Question

Please explain why, I just can't picture it.

Also, what is really the difference between let's say, sin of infinity, and the limit as x approaches infinity of sin (x)

Answer 1

Sine of infinity does not exist. If it were to exist, it would be the limit x to infinity of sin(x) and that does not exist as well.

For a limit to exist, as x gets very big the function must approach and stay near its limit. In the case of sine, it goes between -1 and 1 so there is no limit.

Answer 2

None of those values are defined.

First off, the difference between "sin of infinity" and "the limit as x approaches infinity of sin (x)" is that "sin of infinity" is completely meaningless, while "the limit..." is something we can at least try to evaluate. It's just not meaningful to talk about the value of a function at infinity, because infinity isn't an actual value. The limit, on the other hand, characterizes the behavior of the function at extremely high values.

None of these values are defined because trigonometric functions are periodic. Therefore, they do not approach a particular value as their argument approaches infinity. They continue to cycle through their entire ranges with no change in amplitude or frequency.

Answer 3

All periodic functions, including sin, cos and tan, have no limit as x approaches infinity because of their oscillation behaviors

Answer 4

These are trigonometric functions of angles, based on a right-angled triangle including the given angle. Since there can be no such thing as an angle of infinity, there can be no functions of infinity. Since you cannot have an obtuse angle in a right-angled triangle, there can be no functions of angles greater than 90 degrees. The range of sine values (side opposite/hypotenuse) goes from zero (for zero degrees) to one (for 90 degrees); the range of cosine values (side adjacent/hypotenuse) goes from one (for zero degrees) to zero (for 90 degrees). The range of tangent values (side opposite/side adjacent) goes from zero (for zero degrees) to infinity (for 90 degrees). That is the only reference to "infinity" among the three basic trig functions. The extreme values of each of the functions are actually limits, since you cannot have a triangle with a zero degree angle or one with two 90 degree angles.