- #1
Alexx1
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Can someone prove this?
For all x∈]0;2pi[ : tan(x) > x
For all x∈]0;2pi[ : tan(x) > x
Mark44 said:No, because it isn't true on the interval you gave. The tangent function is undefined at x = pi/2 and x = 3pi/2. For all x in (pi/2, pi), tan(x) < x, and the same is true for the interval (3pi/2, 2pi).
HallsofIvy said:But [itex]\pi/2[/itex] and [itex]3\pi/2[/itex] do belong to that interval! That's what Mark44 was saying. And, in fact, for [itex]\pi/2< x< \pi[/itex], tan(x) is negative and can't possibly be larger than x!
You need to reduce to [itex]]0, \pi/2[[/itex] in order to have tan(x)< x.
Look at the function f(x)= tan(x)- x. It's derivative is sec^{2}(x)- 1 and since sec(x)> 1 for all x in [itex]]0, \pi/2[[/itex], that derivative is positive.
Mark44 said:No, because it isn't true on the interval you gave. The tangent function is undefined at x = pi/2 and x = 3pi/2. For all x in (pi/2, pi), tan(x) < x, and the same is true for the interval (3pi/2, 2pi).
The inequality "tan(x) > x" means that the tangent of x is greater than x for all values of x that are between 0 and 2pi (excluding 0 and 2pi). In other words, the tangent function is increasing on this interval, and the value of the tangent function at any given point is always greater than the value of x at that point.
This inequality can be proven using the properties of the tangent function and the interval given. One approach is to use the derivative of the tangent function, which is sec^2(x), to show that the function is increasing on the given interval. Another approach is to use the geometric definition of the tangent function and the properties of triangles to show that the ratio of the opposite side to the adjacent side is always greater than the angle itself.
No, this inequality is only true for values of x that are between 0 and 2pi, excluding 0 and 2pi. This is because the tangent function has a period of pi, and any values outside of this interval can be converted to values within the interval by adding or subtracting multiples of pi.
The inequality "tan(x) > x" means that the graph of the tangent function is always above the line y=x for values of x between 0 and 2pi (excluding 0 and 2pi). This can be seen on the graph of the tangent function, where the curve of the function is always above the line y=x within this interval.
Yes, this inequality can be used to solve other trigonometric equations by applying the same techniques used to prove it. By understanding the properties of the tangent function and the given interval, we can use this inequality to make conclusions about other trigonometric functions and their relationships with angles in a similar interval.