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 &gt; 0 , y &gt; 0 = arctan ⁡ ( 1 − 1 3 1 9 1 3 + 1 9 ) = arctan ⁡ ( 2 ) So it is arctan ⁡ ( 2 ) + arctan ⁡ ( 7 19 ) .

Question

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        &amp;gt;        0        ,        y        &amp;gt;        0                                          =        arctan        ⁡        (                              1            −                          1              3                                      1              9                                                          1              3                        +                          1              9                                      )                                          =        arctan        ⁡        (        2        )            So it is   arctan  ⁡  (  2  )  +  arctan  ⁡  (      7    19    )  .

meexeniexia17h · Accepted Answer

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 · Accepted Answer

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          ∘

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

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