Just a bit of trigonometry!

Algebra Level 2

What is the numeric value of cot ⁡ ( 4 tan ⁡ − 1 ( 1 5 ) − π 4 ) \cot \left(4 \tan ^{-1}\left(\frac{1}{5}\right)-\frac{\pi }{4}\right) ?


The answer is 239.

A Former Brilliant Member by A Former Brilliant Member

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

3 solutions

Chew-Seong Cheong
Apr 30, 2019

cot ⁡ ( 4 tan ⁡ − 1 1 5 − π 4 ) = tan ⁡ ( π 2 − 4 tan ⁡ − 1 1 5 + π 4 ) Note that cot ⁡ θ = tan ⁡ ( π 2 − θ ) = tan ⁡ ( 3 π 4 − 4 tan ⁡ − 1 1 5 ) and tan ⁡ ( π − θ ) = − tan ⁡ θ = − tan ⁡ ( 4 tan ⁡ − 1 1 5 + π 4 ) Let x = tan ⁡ − 1 1 5 = tan ⁡ 4 x + 1 tan ⁡ 4 x − 1 Since tan ⁡ 2 θ = 2 tan ⁡ θ 1 − tan ⁡ 2 θ = 2 tan ⁡ 2 x + 1 − tan ⁡ 2 2 x 2 tan ⁡ 2 x − 1 + tan ⁡ 2 2 x Divide up and down by tan ⁡ 2 x = 2 + cot ⁡ 2 x − tan ⁡ 2 x 2 − cot ⁡ 2 x + tan ⁡ 2 x Note that tan ⁡ 2 x = 2 × 1 5 1 − 1 25 = 5 12 = 2 + 12 5 − 5 12 2 − 12 5 + 5 12 Multiply up and down by 60 = 120 + 144 − 25 120 − 144 + 25 = 239 \begin{aligned} \cot \left(4 \tan^{-1}\frac 15 - \frac \pi 4\right) & = \tan \left(\frac \pi 2 - 4 \tan^{-1}\frac 15 + \frac \pi 4\right) & \small \color{#3D99F6} \text{Note that }\cot \theta = \tan \left(\frac \pi 2 - \theta\right) \\ & = \tan \left(\frac {3\pi} 4 - 4 \tan^{-1}\frac 15 \right) & \small \color{#3D99F6} \text{and } \tan (\pi - \theta) = - \tan \theta \\ & = - \tan \left(4 {\color{#3D99F6} \tan^{-1}\frac 15} + \frac \pi 4 \right) & \small \color{#3D99F6} \text{Let } x = \tan^{-1} \frac 15 \\ & = \frac {\tan 4x + 1}{\tan 4x - 1} & \small \color{#3D99F6} \text{Since } \tan 2 \theta = \frac {2\tan \theta}{1-\tan^2 \theta} \\ & = \frac {2\tan 2x + 1 - \tan^2 2x }{2\tan 2x - 1 + \tan^2 2x} & \small \color{#3D99F6} \text{Divide up and down by } \tan 2x \\ & = \frac {2 + \cot 2x - \tan 2x }{2 - \cot 2x + \tan 2x} & \small \color{#3D99F6} \text{Note that } \tan 2x = \frac {2\times \frac 15}{1- \frac 1{25}} = \frac 5{12} \\ & = \frac {2+\frac {12}5- \frac 5{12}}{2-\frac {12}5+ \frac 5{12}} & \small \color{#3D99F6} \text{Multiply up and down by } 60 \\ & = \frac {120+144-25}{120-144+25} \\ & = \boxed{239} \end{aligned}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

4ATan(1/5)=ATan(120/119). Therefore Cot(4ATan(1/5)-π/4) =[1+Cot(4ATan(1/5))]/[1-Cot(4ATan(1/5))]=239

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Correct. This also was discussed in the encyclopedia article from which this formula was derived.

A Former Brilliant Member - 2 years, 1 month ago

Machin's Formula.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...