how do i prove that Tan y = y when y is small?

Question

i know that you memorize them but how do i prove this statement?

Answer 1

tan y = sin y / cos y

when y is small, sin y is also very small, but cos y is very big (very close to one)

When you divide small number by 1 you get very small number (but I'm not sure how close that number will be to y).

Answer 2

There are two methods that i know but there is many more methods! You can replace the variable to y if you want but i like x better! and the assumption is x is small.

Method 1: Using the knowledge that sin x= x and cos x= 1-0.5 x^2
we use tan x= sin x/ cos x = x/ (1-0.5 x^2) = x(1-0.5 x^2)^(-1) = x(1+ 0.5x^2+ ... ...) = x + 0.5 x^3 +... ... =x as higher powers of x are negligible!

Method 2: using Maclaurin's expansion!
y= tan x
y'= (sec x)^2
.
.
.(i think you should be able to work it out yourself right?)

then sub x=0, y= 0
y'= 1
y''=...
...(but powers higher than one can be neglected)
so tan x = x

From the formula list, the Maclaurin expansion of tan(x) is
1 3 2 5 17 7 62 9 10
tan(x) = x + --- x + --- x + ---- x + ----- x + O(x )
3 15 315 2835

so actually you can deduce from formula list too if you want!

opss the number is a little out of place but the first number is for the fraction and the second is for the power of x for the second fraction and so on
YAY!!

Answer 3

One way to prove or more correctly to demonstrate is to use Taylor's theorem. Do a series expansion for tan y.

Taking the limit of the series as y --> 0 will give you tan y ~ y.

The expansion will be:

tan y = y + 2y^3/3! + 16y^5/5! + 272y^7/7! + ....

Answer 4

Actually as stated this is not true. It is true for 0 (tan(0) = 0), and it is true that lim y->0 tany/y = 1. I believe it is this last one that you want to prove.

tany/y = siny/(ycosy)

lim y->0 siny/(ycosy) = (lim siny/y)*(lim 1/cosy)

The right term is easy, since 1/cosy is continuous and defined at y = 0 it is just 1.

For the left there are a few ways to evaluate lim siny/y. One is if you already know this is 1.

Another is L'Hospital rule, taking the derivative of top and bottom and evaluating at 0 (cosy/1)

Answer 5

tan y=y
sin y =y
where, y is in radians not degrees
y is so small

this is proved numerically, because the raltionship between y, and tan y, or sin y or cos y are numerical realtionships. these values were found by accurate measuring to correlate betwwn the angles and lengths.

Answer 6

as y tends to zero the opposite side will tend to zero so tany will tend to zero.so the difference between y and tany will be negligible.so we can say tany=y

Answer 7

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