what is $x$ for $\tan^23x = 2\sin^23x$

Question

If $x$ = acute angle then find $x$ such that $\tan^23x = 2\sin^23x$.

So $\tan3x = \sqrt2\sin3x$

$\frac{1}{\cos3x} = \sqrt2$

$3x = 45^{\circ}$

what are all the possibilies for $x$ ? because the question asked for all possibilities of $x$

The options are: $180, 195, 120, 135, 360$

Answer 1

\begin{align} \tan^23x&=2\sin^23x\\ \tan^23x-2\sin^23x&=0\\ \sin^23x(\frac1{\cos^23x}-2)&=0\\ \sin^23x(\sec^23x-2)&=0 \end{align}

So we have either \begin{align} \sin^23x&=0\\ 3x&=0^{\circ}, 180^{\circ}, 360^{\circ}\\ x&=0^{\circ}, 60^{\circ}, 120^{\circ} \end{align} or \begin{align} \sec^23x-2&=0\\ \sec^23x&=2\\ \sec3x&=\pm\sqrt2\\ 3x&=45^{\circ}, 135^{\circ}, 225^{\circ}, 315^{\circ}\\ x&=15^{\circ}, 45^{\circ}, 75^{\circ}, 105^{\circ} \end{align}

From the above, the only admissible solutions are $\boxed{15^{\circ}, 45^{\circ}, 60^{\circ}, 75^{\circ}}$.

The sum of these angles is $\boxed{195^{\circ}}$.

Answer 2

Here's a plot that should help:

Answer 3

Do some trigonometry and find the general solution first: $$\tan^23x=2\sin^23x=\frac{2\tan^23x}{1+\tan^23x}\iff\tan^23x(1+\tan^23x)=2\tan^23x$$ So,

• either $\tan3x=0\iff 3x\equiv 0\mod180\iff x\equiv 0\mod 60$;
• or $\tan^23x=1\iff\tan 3x=\pm 1\iff 3x\equiv \pm45\mod 180\iff x\equiv\pm 15\mod 60$.

Then select the values in $\bigl[0,90]$.

Answer 4

If $\cos6x=a,$

Using $\cos2y=1-2\sin^2y=\dfrac{1-\tan^2y}{1+\tan^2y}$

$$\dfrac{1-a}{1+a}=1-a$$

$$\implies a(a-1)=0$$

If $\cos6x=0,6x=(2n+1)90^\circ$ where $n$ is an integer

If $\cos6x=1,6x=360^\circ m$