Find arctan(1/3)+arctan(1/9)+arctan(7/19)

odcizit49o

odcizit49o

Answered question

2022-11-07

Find arctan ( 1 3 ) + arctan ( 1 9 ) + arctan ( 7 19 )
Firstly used this formula
arctan ( α ) + arctan ( β ) = arctan ( 1 x y x + y ) , x > 0 , y > 0 = arctan ( 1 1 3 1 9 1 3 + 1 9 ) = arctan ( 2 )
So it is arctan ( 2 ) + arctan ( 7 19 ) .

Answer & Explanation

meexeniexia17h

meexeniexia17h

Beginner2022-11-08Added 18 answers

The correct formula is:
arctan ( u ) + arctan ( v ) = arctan ( u + v 1 u v )
So:
arctan ( 1 / 3 ) + arctan ( 1 / 9 ) = arctan ( 6 13 )
arctan ( 1 / 3 ) + arctan ( 1 / 9 ) + arctan ( 7 / 19 ) = arctan ( 6 / 13 + 7 / 19 1 6 / 13 7 / 19 )
= arctan ( 1 ) = π 4
kaltEvallwsr

kaltEvallwsr

Beginner2022-11-09Added 8 answers

Just calculate:
tan ( arctan 1 3 + arctan 1 9 + arctan 7 19 ) =
= 1 3 + 1 9 1 1 3 1 9 + 7 19 1 1 3 + 1 9 1 1 3 1 9 7 19 = 1 ,
which gives the answer: 45

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