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solve tanx=x (1 Viewer)
- Thread starter Motvoo
- Start date
Pwnage101
Moderator
There is no solution in the given interval.
You could sketch both functions tan(x) and x on the same axes and see why.
A more algebraic approach could include:
Clearly x=0 is a solution, since tan(0)=0.
Now, let f(x)=tanx-x
---> f'(x)=[sec(x)]^2-1
Now, for 0<x<pi /2, we know 0<cos(x)<1, so 0 < [cos(x)]^2 < 1, hence [sec(x)]^2>1.
Hence f'(x)=[sec(x)]^2-1>1-1=0 for 0<x<pi/2.
Now f(0)=0, and f(x) is a strictly increasing function [f'(x)>0] when 0<x<pi/2, hence for 0<x<pi/2, f(x)>0. Hence f(x) does NOT equal 0 when 0<x<pi/2.
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If we use a graphical method... sketch y = x and y = tanx
If you sketch to scale, you will see that the curves intersect at 0.
Noting that tan (x) > x for [0 < x < pi/2... this is can be easily proven], the next solution will be in between (pi)/2 and 3(pi)/2 and hence out of the domain given in the question.
Therefore, for the domain 0 < x < pi/2, there would be no solution to tan(x)=x
... there should be a more elegant method though ^^
If you wanted to find the other solutions outside the given domain, try letting f (x) = tanx – x, estimating the x- ordinate close to a solution on the graph and use Newton’s method
If you sketch to scale, you will see that the curves intersect at 0.
Noting that tan (x) > x for [0 < x < pi/2... this is can be easily proven], the next solution will be in between (pi)/2 and 3(pi)/2 and hence out of the domain given in the question.
Therefore, for the domain 0 < x < pi/2, there would be no solution to tan(x)=x
... there should be a more elegant method though ^^
If you wanted to find the other solutions outside the given domain, try letting f (x) = tanx – x, estimating the x- ordinate close to a solution on the graph and use Newton’s method
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Pwnage101
Moderator
I am of the opinion that "the more solutions, the better" with regards to solving problems in mathematics in different ways. It really helps you get used to various different methods, and in doing so you learn new techniques, so don't worry whether or not someone has already posted, your input is valuable.
A good example is my post in this thread.
Thanks and congratulations on your ATAR!
Thanks!
hscishard
Active Member
More solutions? You mean ways to solve a question...because there are fixed number of answers for a question
Ohhh..i should read everything first
Ohhh..i should read everything first