Tangential Progression 1

Geometry Level 2

$\large \tan(50^\circ) \tan(60^\circ) \tan(70^\circ) = \ ?$

\tan(85^\circ)

\tan(80^\circ)

\tan(75^\circ)

\tan(77.5^\circ)

1 solution

Brian Charlesworth
Aug 4, 2015

We can make use of the identity $\tan(x)\tan(60^{\circ} - x)\tan(60^{\circ} + x) = \tan(3x).$

Letting $x = 10^{\circ},$ we then have that

$\tan(10^{\circ})\tan(50^{\circ})\tan(70^{\circ}) = \tan(30^{\circ})$

$\Longrightarrow \tan(50^{\circ})\tan(60^{\circ})\tan(70^{\circ}) = \boxed{\tan(80^{\circ})},$

since $\tan(\theta) = \dfrac{1}{\tan(90^{\circ} - \theta)}.$

Yes! This is how I've done it. I started by noticing that $50 + 60 + 70 = 180$ , and I thought of applying $\tan(A) + \tan(B) + \tan(C) = \tan(A) \tan(B) \tan(C)$ but that requires more work to obtain the final answer of $\tan(80^\circ)$ .

Try my other question too ! Pretend this question does not exist lol

Chung Kevin - 5 years, 10 months ago

Haha I actually solved the other problem first, noticing that $\tan(20)\tan(30)\tan(40)$ was the reciprocal of $\tan(50)\tan(60)\tan(70)$ and then solving in the same manner as above. So when I saw this problem I was all set. :)

Brian Charlesworth - 5 years, 10 months ago

Tangential Progression 1

1 solution

0 pending reports