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In an acute triangle $x,y,z$ are the given angles where $\cos x=\tan y$, $\cos y = \tan z$ and $\cos z = \tan x$. Find the sum of sines in the triangle.

Could be done by substituting values of $\sin$ function but in vain. Can anyone please help me?

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    $\begingroup$ Normally you should show your efforts so far. $\endgroup$
    – Ripi2
    May 19, 2020 at 19:29
  • $\begingroup$ The sum of sines in the triangle does not have any sense, if such triangle does not exist. $\endgroup$
    – g.kov
    May 20, 2020 at 3:47

2 Answers 2

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Hint:

Let $a=\tan^2 x, b =\tan^2y, c=\tan^2 z$. By squaring and using $\cos^2x =\frac{1}{1+\tan^2x}$, we have $$ b=\frac{1}{1+a} \\ c=\frac{1}{1+b} \\ a=\frac{1}{1+c}$$ By substituting $c$ and $b$ you get a quadratic in $a$ : $$a^2+a-1=0 \implies a=\frac{\sqrt 5-1}{2}$$ and from here you get $$\sin x =\sqrt{\frac{\sqrt 5-1}{\sqrt 5+1}}$$ by some simple trig identities. You can find $\sin y$ and $\sin z$ in a similar way.

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  • $\begingroup$ The hint is nice, but the solution is (wrong) incomplete. The answer implies that $\tan x=\tan y=\tan z=\sqrt{\tfrac12\,(\sqrt 5-1)}$. What kind of a triangle is that? $\endgroup$
    – g.kov
    May 20, 2020 at 3:39
  • $\begingroup$ @g.kov As the question, so the answer. $\endgroup$
    – Tavish
    May 20, 2020 at 8:29
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The question states that it's given that in an acute triangle $x,y,z$ are the given angles where

\begin{align} \cos x&=\tan y \tag{1}\label{1} ,\\ \cos y&=\tan z \tag{2}\label{2} ,\\ \cos z&=\tan x \tag{3}\label{3} . \end{align}

There are many ways to prove that there is no valid triangle with such properties.

For one is the incomplete answer which implies that it follows from \eqref{1}-\eqref{3} that \begin{align} \tan x&=\tan y=\tan z= \cos x=\cos y=\cos z= =\sqrt{\tfrac12\,(\sqrt 5-1)} , \end{align}

which is absurd.

Another way is: rewriting \eqref{1}-\eqref{3} as

\begin{align} \cos x\cos y&=\sin y \tag{4}\label{4} ,\\ \cos y\cos z&=\sin z \tag{5}\label{5} ,\\ \cos z\cos x&=\sin x \tag{6}\label{6} , \end{align}
so \begin{align} \sin x+\sin y+\sin z &= \cos x\cos y+\cos y\cos z+\cos z\cos x \tag{7}\label{7} . \end{align}

Using known identities

\begin{align} \sin x+\sin y+\sin z&=u \tag{8}\label{8} \end{align}

and

\begin{align} \cos x\cos y+\cos y\cos z+\cos z\cos x &=\frac{u^2+v^2}4-1 \tag{9}\label{9} , \end{align}

where $u=\rho/R$, $v=r/R$ and $\rho,r,R$ are the semiperimeter, inradius and circumradius of given triangle (if such exists).

From equations \eqref{7}-\eqref{9},

\begin{align} \frac{u^2+v^2}4-1=v ,\\ u&=2+\sqrt{8-v^2} , \end{align}

and this expression for

\begin{align} u>\tfrac{3\sqrt3}2 &=\max_{v\in[0,1/2]}u(v) , \end{align}

that is, there is no a pair $(u,v)$ that simultaneously agree with \eqref{7}-\eqref{9} and represent a valid triangle.

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