Horizontal tangents to sin2x-2sinx?

Question

I know that the horizontal tangent lines are when the derivative is equal to 0, and for the derivative I got 2cos2x-2cosx. But when I set this to zero I can't figure out how to get rid of the cos2x and make it in terms of x...like what identity to use so I can factor it and whatnot.

Thanks!

Answer 1

There's a trig identity that states:
cos(2x) = 2cos(x)^2 - 1
So 2cos(2x) = 4cos(x)^2 - 2.

2cos(2x) - 2cosx = 4cos(x)^2 - 2 - 2cosx

For more ease in factoring the polynomial, let y = cosx for now. Then the polynomial is:
4y^2 - 2 - 2y.

Rearrange and factor out a 2:

2 * (2y^2 - y -1)

Looks more factorable. (2y^2 - y - 1) can be factored as
(2y + 1)*(y - 1).

So y = -0.5 or 1.

Remember, though, that y = cosx. So cosx = -0.5 or 1.

Solve for x.

Answer 2

At a point with horizontal tangent the derivative must be zero
So f´= 2cos2x-2cosx.But cos2x = cos^2(x) -sin^2(x) =

= 2cos^2(x)-1.So your derivative becomes 4cos^2(x) -2cosx-2 =0

Call cos(x)= z so 2z^2 -z-1=0 and z= (1+-3)/4

z= 1 and z= -1/2 cos x= 1 x=2kpi

cos x=-1/2 x=2pi/3+2kpi and x= 4pi/3 +2kpi

(k is a positve or negative integer including 0)

Answer 3

Your derivative is correct. Use cos2x = 2(cosx)^2 - 1 to turn this into a quadratic expression in cosx. When you put this equal to zero you should be able to solve the resulting quadratic equation.

Answer 4

f ` (x) = 2cos 2x - 2 cos x = 0 for horizontal tangent

cos 2x - cos x = 0

But cos 2x = 2cos² x - 1 so:-

2cos²x - 1 - cos x = 0

2cos²x - cosx - 1 = 0

(2cosx + 1).(cosx - 1) = 0

cosx = -1/2, cos x = 1

x = 120° , 240° , 0° , 360°

Answer 5

cos2x=2(cosx)^2-1
=1-2(sinx)^2

Answer 6

The graph crosses on the factors 0, 3, -3, 6, +6, and so on. only save on with th development. in case you have a graphing calculator (probability is, you do!) only plug the equation directy into your graph, (make advantageous in "Mode" button, its in Radians, no longer tiers, yet remem. to alter it lower back to tiers when you're executed with the graph. wish this facilitates! :)