Question

What is the angle between the graph of `f(x)=2sin(x)-1` and the `x`-axis if the graph of `f` intersects the `x`-axis at `x=pi/6`?

I worked out the tangent line to the graph of `f` at `(pi/6, 0)` and I got `y=2cos(x)*x-pi/6`

Answer 1

The angle is `=pi/3`

Explanation:

We need

`(sinx)'=cosx`

`(a)'=0`

Our function is

`f(x)=2sinx-1`

The derivative with respect to `x` is

`f'(x)=(2sinx-1)'==(2sinx)'-(1)'=2cosx`

At the point `x=pi/6`,

`f'(pi/6)=2cos(pi/6)=2*sqrt3/2=sqrt3`

The slope of the tangent at `x=pi/6` is

`m=sqrt3`

The equation of the tangent is

`y=sqrt3(x-pi/6)`

`m=tanalpha=sqrt3`

Therefore,

The angle is `alpha=arctan(sqrt3)=60^@=pi/3`

graph{((2sinx-1))(y-(sqrt3)(x-pi/6))=0 [-0.915, 1.786, -0.586, 0.765]}