Solving sin2x = 3cosx for all values of x

[Jen123]

solve sin2x = 3cosx for all values of x. answer in radians

This is what I got 14pi/45 + pi(k) is this correct
sin2x / cosx = 3
tan2x=3
tanx = 3/2
[(14pi) / (45) ] + (pi)k

 

[pka]

There are many solutions: $\displaystyle \frac{{\left( {2k + 1} \right)}}{2}\pi .$

 

[Jen123]

can you type out how you got the answer

 

[galactus]

\(\displaystyle \L\\\frac{sin(2x)}{cos(x)}\neq{tan(2x)}\)...wrong

\(\displaystyle \L\\\frac{sin(2x)}{cos(x)}=2sin(x)\)...correct

 

[skeeter]

sin(2x) = 3cosx

2sinxcosx = 3cosx

2sinxcosx - 3cosx = 0

cosx(2sinx - 3) = 0

cosx = 0 or sinx = 3/2

since -1 < sinx < 1, sinx = 3/2 has no solution.

cosx = 0 at every odd multiple of pi/2.

 

[pka]

\(\displaystyle \begin{array}{l}
\sin (2x) = 3\cos (x) \\
2\sin (x)\cos (x) = 3\cos (x) \\
2\sin (x) = 3\quad \left[ {\cos (x) \not= 0,\quad x \not= \frac{{\left( {2k + 1} \right)\pi }}{2}} \right] \\
\end{array}\)
So there are no other slutions.

 

[Mrspi]

Re: Solving for all values of x

solve sin2x = 3cosx for all values of x. answer in radians

This is what I got 14pi/45 + pi(k) is this correct
sin2x / cosx = 3
tan2x=3
tanx = 3/2
[(14pi) / (45) ] + (pi)k

I'm not at all sure what you've done here.

I'd approach it this way.

sin 2x = 2 sin x cos x

So,

2 sin x cos x = 3 cos x

Subtract 3 cos x from both sides:

2 sin x cos x - 3 cos x = 0

Factor cos x out of both terms:

cos x (2 sin x - 3) = 0

If the product of two factors is 0, then at least one of the factors must have a value of 0.

If cos x = 0, then x must be an odd multiple of pi/2 (because it must terminate on the y-axis).
x = (2n + 1)*(pi/2) for any integer value of n.

If 2 sin x - 3 = 0
then 2 sin x = 3
and sin x = 3/2
There are no values of x which would make this true (the maximum value of sin x is 1).

I hope this helps you.

 

[morson]

Re: Solving for all values of x

and sin x = 3/2
There are no values of x which would make this true

'cept \(\displaystyle \L\ x = \frac{\pi}{2}\ (4k + 1) + i ln(\frac{3 - \sqrt{5}}{2}\)\)

for integral k. Kinda frivolous, but there if they want ALL values of x.