Math and Arithmetic
Trigonometry

# How do you solve cosx - xsinx equals 0?

To solve this, you need to find values of x where cos(x)

=

xsin(x).

First of all, 0 is not a solution because cos(0) =

1, and sin(0) =

0. Since 0 is not a solution, divide both sides of the equation by sin(x)

to get cot(x)

=

x (remember that cos divided by sin is the same as cot). The new question to answer is, when is cot(x)

=

x? Using Wolfram Alpha, the results are

x Â±9.52933440536196...
x Â±6.43729817917195...
x Â±3.42561845948173...
x Â±0.860333589019380... there will be an infinite number of solutions.

If you'd like to do the calculation yourself (not asking WolframAlpha)

then there's a trick which almost always works, even for equations which cant be done analytically.

Starting with the basic equation, cos(x)

=

x*sin(x),

transpose it to a form starting with "x =".

In this case you could get: x =

1/tan(x), x =

cot(x)

or from tan(x)

=

1/x you get x =

Arctan(1/x).

Because I like to do my calcs

on an old calculator which only has Arctan

and not Arccotan

(Inverse cotangent(x))

I use the last above - x =

Arctan(1/x)

Starting with a value like 0.5, hit the 1/x key then shift tan keys. Just keep repeating those two operations and the display will converge on 0.860333. Too easy. This example of the method is not a good one as it takes about 25 iterations to converge to within 0.0000001 of the right answer. It is unusually slow.

And finally, this method has only 50% chance of working first try. We were lucky picking x =

Arctan(1/x). x =

1/tan(x) diverges ind the iterations do not converge on the answer.

So if you try this method on another problem and it diverges, just transpose the equation again and have another go.

Starting with x^2 + x - 3 =

0,

and iterating x =

3-x^2, you find it diverges, so

try x =

sqr(3-x) which (with care and about 25 iterations) converges on 1.302775638.

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## Related Questions

2sinxcosx-cosx=0 Factored : cosx(2sinx-1)=0 2 solutions: cosx=0 or sinx=.5 For cosx=0, x=90 or 270 degrees For sinx=.5, x=30 degrees x = {30, 90, 270}

2cos2x - cosx -1 = 0 Factor: (2cosx + 1)(cosx - 1) = 0 cosx = {-.5, 1} x = {...0, 120, 240, 360,...} degrees

To clarfiy, is there anyway I can solve a system of equations such as: Asin(x) + Bcos(y) - D = 0 Bcos(x) + Asin(y) - C = 0

cosx + sinx = 0 when sinx = -cosx. By dividing both sides by cosx you get: sinx/cosx = -1 tanx = -1 The values where tanx = -1 are 3pi/4, 7pi/4, etc. Those are equivalent to 135 degrees, 315 degrees, etc.

secx = 1/cosxand 1/cotx = tanx, therefore1/cosx + tanx = 1 + sinx/cosx, andsin/cos = tanx, therefore1/cosx + tanx = 1 + tanx, therefore1/cosx = 1, therfore1 = cosx.So, therfore, it is not neccesarily true.But if you meansecx plus 1 divided by cotx equals (1 plus sinx) divided by cosx(this is probably what you mean) Let's start over!secx = 1/cosxand 1/cotx = tanx, therefore1/cosx + tanx = (1+sinx)/cosx therefore1/cosx + tanx = 1/cosx + sinx/cosxsinx/cosx = tanx therfore1/cosx + tanx = 1/cosx + tanxDo you think this is correct? Subtract both sides by 1/cosx + tanx:0 = 0So, therefore, this is correct!(BTW, I'm in Grade 6! :P)

X=60 how did you get that? could you show all the steps?

You will have to bear with the angle being represented by x because this browser will not allow characters from other alphabets!sin^2x + cos^2x = 1=> sin^2x = 1 - cos^x = (1 + cosx)(1 - cosx)Divide both sides by sinx (assuming that sinx is not zero).=> sinx = (1 + cosx)(1 - cosx)/sinxDivide both sides by (1 - cosx)=> sinx/(1 - cosx) = (1 + cosx)/sinx=> sinx/(1 - cosx) - (1 + cosx)/sinx = 0

cos x - 0.5 = 0 &rArr; cos x = 0.5 &rArr; x = 2n&pi; &plusmn; &pi;/3

"x equals 0" is an equality, not an inequality. The question is, therefore, not consistent.

Suppose x3-4x = 0. To solve, factor: x3-4x = x(x2-4) = x(x+2)(x-2) = 0 Now, a product equals 0 if and only one or more of the factors equals 0, so set each factor to 0 and solve. The roots are 0,-2 and +2.

tanx=2cscx sinx/cosx=2/sinx sin2x/cosx=2 sin2x=2cosx 1-cos2x=2cosx 0=cos2x+2cosx-1 Quadratic formula: cosx=(-2&plusmn;&radic;(2^2+4))/2 cosx=(-2&plusmn;&radic;8)/2 cosx=(-2&plusmn;2&radic;2)/2 cosx=-1&plusmn;&radic;2 cosx=approximately -2.41 or approximately 0.41. Since the range of the cosine function is [-1,1], only approx. 0.41 works. So: cosx= approx. 0.41 Need calculator now (I went as far as I could without one!) x=approx 1.148

a2+30a+56=0 , solve for a Using the quadratic formula, you will find that: a=-2 , a=-28

Yes, that looks good. That's 180 degrees plus every multiple of 360 degrees more.

4x2+50x = 0 4x(x+12.5) = 0 x = 0 or x = -12.5

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