If $2\sin\theta+\cos\theta=\sqrt3$, what is the value of $\tan^2\theta+4\tan\theta$ ?


First I tried plugging in some values for $\theta$ like $0,\frac{\pi}4,\frac{\pi}3,...$ but neither of these known angles worked. But by doing it I realized that for $\theta=\frac{\pi}4+k\pi\quad$, $\tan^2\theta+4\tan\theta=5\quad$ and $2\sin\theta+\cos\theta\neq\sqrt3$ Hence the fourth choice is wrong.

Also tried to expanding, $$\tan^2\theta+4\tan\theta=\dfrac{\sin^2\theta}{\cos^2\theta}+\dfrac{4\sin\theta}{\cos\theta}=\dfrac{\sin^2\theta+4\sin\theta\cos\theta}{\cos^2\theta}$$But can't continue even writing $4\sin\theta\cos\theta=2\sin2\theta$ doesn't help.

  • 2
    $\begingroup$ Add $\frac{\cos^2\theta}{\cos^2\theta}$ to the fraction. Square $2\sin\theta+\cos\theta$. Does that help? $\endgroup$ Jun 19, 2021 at 19:00
  • $\begingroup$ @DonThousand Yes I got it now, thanks! $\endgroup$
    – Etemon
    Jun 19, 2021 at 19:15
  • $\begingroup$ I was wondering if it's possible to get to $\tan(\theta)+2=\sqrt{6}$ as an alternative solution, but I cannot see a clear way to get to that, other than what's have already been proposed... $\endgroup$
    – zwim
    Jun 19, 2021 at 19:40

3 Answers 3


$(2\sin\theta+\cos\theta)^2 = 3$

$4\sin^2\theta + \cos^2\theta + 4 \sin\theta \cos\theta = 3 \sin^2\theta + 3\cos^2\theta$

$\sin^2\theta + 4\sin\theta \cos\theta = 2\cos^2\theta$

Now dividing both sides by $\cos^2\theta$,

$\tan^2\theta + 4\tan\theta = 2$


Alternative: We have $$2\sin\theta=\sqrt3 -\cos \theta$$ Squaring, $$4-4\cos^2 \theta=3+\cos^2\theta-2\sqrt 3 \cos\theta$$ So, $$5\cos^2 \theta-2\sqrt 3 \cos\theta-1=0$$ Or $$\cos \theta=\frac {2\sqrt 3 \pm 4\sqrt 2}{10}$$ Since further conditions haven't been given, both values would be valid, and from here $\tan \theta$ can be calculated.


Simply square the constraint and you get $4 \sin^2(x)+4\sin(x)\cos(x)+\cos^2(x)=3$ then double angle and trig identity $3\sin^2(x)+2\sin(2x)=2$ furthermore this is equivalent to $2\sin(2x) = 3\cos^2(x)-1$

and with your definition of the left hand side


we can insert the definition of $2\sin(2x)$ we got from the constraint and we get


I hope that helps and I hope I didn't make a mistake ;)


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