Find S=(\tan 1^{\circ})+(\tan 2^{\circ})+(\tan 3^{\circ})+(\tan 4^{\circ}) +(\tan 5^{\circ})+(\ta

Anika Klein

Anika Klein

Answered question

2022-01-28

Find S=(tan1)+(tan2)+(tan3)+(tan4)+(tan5)+(tan6o{})+(tan7)+(tan8)+(tan9)+(tan10)
My attempt:
cosx+cos2x+cos3x+...+cosnx=sin(n+12)xsinx22sinx2
sinx+sin2x+sin3x+...+sinnx=cosx2cos(n+12)x2sinx2
Since angles are small we use term by term division which gives:
tanx+tan2x+tan3x+...+tannx=cosx2cos(n+12)xsin(n+12)xsinx2
Putting x=1 and n=10 gives s=0.9629..
Wolfram alpha gives s=0.9653..
Is my calculations correct?

Answer & Explanation

fionaluvsyou0x

fionaluvsyou0x

Beginner2022-01-29Added 11 answers

What you are writing is not correct since, in essence, you write that
ab+cd+ef=a+c+eb+d+f
which is not true. For example
12+34+56=2512while1+3+52+4+6=34
However, since you notice that all angles are small, you could approximate the summation using that, for small x
tan(x)=x+x33+2x515+O(x7)
tan(ix)=ix+i3x33+2i5x515+O(x7)
i=1ntan(ix)=xi=1ni+x33i=1ni3+2x515i=1ni5+
and use Faulhabers

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?