## tan A = tan B + tan C + tan D

$\tan \, 84^\circ \; = \; tan \, 78^\circ \; + \; tan \, 72^\circ \; + \; tan \, 60^\circ$

$\tan \, 80^\circ \; = \; tan \, 70^\circ \; + \; tan \, 60^\circ \; + \; tan \, 50^\circ$

Find other   4-tuples   $(a, b, c, d)$   of distinct integers between   $0^\circ$   and   $90^\circ$   that
satisfy the relation

$tan \, A^\circ\; = \; tan \, B^\circ \; + \; tan \, C^\circ \; + \; tan \, D^\circ$

Other solutions:

$\tan \, 72^\circ \; = \; tan \, 66^\circ \; + \; tan \, 36^\circ \; + \; tan \, 6^\circ$
$\tan \, 70^\circ \; = \; tan \, 60^\circ \; + \; tan \, 40^\circ \; + \; tan \, 10^\circ$
$\tan \, 72^\circ \; = \; tan \, 60^\circ \; + \; tan \, 42^\circ \; + \; tan \, 24^\circ$
$\tan \, 78^\circ \; = \; tan \, 66^\circ \; + \; tan \, 60^\circ \; + \; tan \, 36^\circ$
$\tan \, 78^\circ \; = \; tan \, 72^\circ \; + \; tan \, 42^\circ \; + \; tan \, 36^\circ$

How many such equations can you produce if we allow repeated angles?

Hint:   there are 49 such equations

Determine whether there are 3-term equations and 5-term equations.

math grad - Interest: Number theory
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### 4 Responses to tan A = tan B + tan C + tan D

1. Paul says:

Here are the 4 term ones with distinct integers

Tan 6 + Tan 36 + Tan 66 = Tan 72.
Tan 10 + Tan 40 + Tan 60 = Tan 70.
Tan 24 + Tan 42 + Tan 60 = Tan 72.
Tan 50 + Tan 60 + Tan 70 = Tan 80.
Tan 60 + Tan 72 + Tan 78 = Tan 84.

and those with repeats

Tan 4 + Tan 4 + Tan 43 = Tan 47.
Tan 4 + Tan 82 + Tan 82 = Tan 86.
Tan 5 + Tan 80 + Tan 80 = Tan 85.
Tan 6 + Tan 6 + Tan 42 = Tan 48.
Tan 6 + Tan 36 + Tan 66 = Tan 72.
Tan 6 + Tan 78 + Tan 78 = Tan 84.
Tan 7 + Tan 76 + Tan 76 = Tan 83.
Tan 8 + Tan 8 + Tan 41 = Tan 49.
Tan 8 + Tan 74 + Tan 74 = Tan 82.
Tan 9 + Tan 72 + Tan 72 = Tan 81.
Tan 10 + Tan 10 + Tan 40 = Tan 50.
Tan 10 + Tan 40 + Tan 60 = Tan 70.
Tan 11 + Tan 68 + Tan 68 = Tan 79.
Tan 13 + Tan 64 + Tan 64 = Tan 77.
Tan 14 + Tan 14 + Tan 38 = Tan 52.
Tan 14 + Tan 62 + Tan 62 = Tan 76.
Tan 16 + Tan 58 + Tan 58 = Tan 74.
Tan 17 + Tan 56 + Tan 56 = Tan 73.
Tan 18 + Tan 18 + Tan 36 = Tan 54.
Tan 18 + Tan 54 + Tan 54 = Tan 72.
Tan 19 + Tan 52 + Tan 52 = Tan 71.
Tan 20 + Tan 20 + Tan 35 = Tan 55.
Tan 20 + Tan 50 + Tan 50 = Tan 70.
Tan 22 + Tan 46 + Tan 46 = Tan 68.
Tan 23 + Tan 44 + Tan 44 = Tan 67.
Tan 24 + Tan 42 + Tan 42 = Tan 66.
Tan 24 + Tan 42 + Tan 60 = Tan 72.
Tan 25 + Tan 40 + Tan 40 = Tan 65.
Tan 26 + Tan 38 + Tan 38 = Tan 64.
Tan 27 + Tan 36 + Tan 36 = Tan 63.
Tan 28 + Tan 34 + Tan 34 = Tan 62.
Tan 29 + Tan 32 + Tan 32 = Tan 61.
Tan 50 + Tan 60 + Tan 70 = Tan 80.
Tan 60 + Tan 72 + Tan 78 = Tan 84.

There are no solutions with 2 or 5 terms (0 < Integers <= 90)

Paul.

• benvitalis says:

I also found
$\tan \, 78^\circ \; = \; tan \, 66^\circ \; + \; tan \, 60^\circ \; + \; tan \, 36^\circ$
$\tan \, 78^\circ \; = \; tan \, 72^\circ \; + \; tan \, 42^\circ \; + \; tan \, 36^\circ$
giving us seven 4-term (A, B, C, D) of distinct integers

• benvitalis says:

and those with repeats, I found 49 such equations.

True. There are no such 3-term equations and no such 5-term equations.

2. Ng Ser-Hong says:

For those with repeats indicated by Paul satisfy the relation
Tan x + Tan x + Tan y = Tan (x+y)