Find the points on the graph of y = tan x, 0≤x≤2𝜋, where the tangent line is parallel to the line 4x – y = 7.

Answer 1

For y=tan x where 0≤x≤2𝜋

We know that tan x = sin x/ cosx

So first, let's get the derivative of y using the quotient rule:

y' = (cos x * cos x - (-sin x) sin x)/cos²x

y' = (cos²x + sin²x)/cos²x

y' = 1/cos²x

y' = sec²x

For the line 4x-y=7, let's put it in slope-intercept form (y=mx+b) so we can get the slope (m):

-y=-4x+7

y=4x-7

Therefore the slope m=4

*Remember, two distinct lines are parallel if and only if they have the same slope. Therefore:

sec²x = 4

sec x = ± 2

For sec x = 2

cos x =1/2

x = π/3 , 5π/3

For sec x = -2

cos x = -1/2

x= 2π/3 , 4π/3

Let's get the tangent of those angles:

tan(π/3) = √3, tan(2π/3) = = -√3

tan(4π/3) = √3, tan(5π/3) = = -√3

Therefore, the points that we are looking for are:

(π/3, √3), (2π/3, -√3), (4π/3, √3), (5π/3, -√3)