Trigonometric EquationHard
Question
The smallest positive root of the equation, tan x - x = 0 lies in
Options
A.
B.
C.
D.
Solution
Let f(x) = tan x - x
we know, for 0 < x <
⇒ tan x > x
∴ f(x) = tan x - x has no root in (0, π / 2)
For π / 2 < x < π, tan x is negative.
∴ f(x) = tan x - x < 0
So f(x) = 0 has no root in
For
< x < 2π tan x is negative
∴ f(x) = tan x - x < 0
So, f(x) = 0 has no root in
We have, f(π) = 0 - π < 0
and
∴ f(x) = 0 has at least one root between π and
we know, for 0 < x <
⇒ tan x > x
∴ f(x) = tan x - x has no root in (0, π / 2)
For π / 2 < x < π, tan x is negative.
∴ f(x) = tan x - x < 0
So f(x) = 0 has no root in
For
< x < 2π tan x is negative ∴ f(x) = tan x - x < 0
So, f(x) = 0 has no root in
We have, f(π) = 0 - π < 0
and
∴ f(x) = 0 has at least one root between π and
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