Question #ba28d

1 Answer
Feb 13, 2018

x=0.86 radians =49.02^@, and

x=pi-0.86=2.28 radians =180^@-49.02^@=130.98^@

Explanation:

.

cot^3x/sinx=cotx

Let's separate the numerator of the left hand side into two pieces:

cot^3x=cotx(cot^2x)

cotx(cot^2x/sinx)=cotx

Let's divide both sides by cotx:

cancelcolor(red)cotx(cot^2x/sinx)=cancelcolor(red)cotx

cot^2x/sinx=1

Let's multiply both sides by sinx:

sinx(cot^2x/sinx)=sinx

cancelcolor(red)sinx(cot^2x/cancelcolor(red)sinx)=sinx

cot^2x=sinx

But we know that:

cotx=cosx/sinx

Let's plug this in:

cos^2x/sin^2x=sinx

Let's multiply both sides by sin^2x:

sin^2x(cos^2x/sin^2x)=sin^3x

cancelcolor(red)(sin^2x)(cos^2x/cancelcolor(red)(sin^2x))=sin^3x

cos^2x=sin^3x

We know that:

sin^2x+cos^2x=1 :. cos^2x=1-sin^2x

Let's plug this in because, by doing this, we can turn the equation into one that has all its terms in the form of the same variable:

1-sin^2x=sin^3x

sin^3x+sin^2x-1=0

This is in the form of a polynomial of degree 3. There are various methods for solving an equation like this. You can either use a graphing utility, or rational roots theorem, or the Newton-Raphson method.

The last one is typically covered in calculus. If you have not taken calculus you can use one of the first two methods. Let's use a graphing utility in this case.

We can say:

z=sinx and write the equation as:

z^3+z^2-1=0

The graph is:

enter image source here

The graph crosses the x-axis at:

z=0.75488

This means:

sinx=0.75488

x=arcsin(0.75488)

x=0.86 radians =49.02^@

and

x=pi-0.86=2.28 radians =180^@-49.02^@=130.98^@