Trigonometry equation problemShow that arctan ⁡ 2 = arccos ⁡ ( 1 / 5 ) .This is the result of solving one trigonometrical equation.

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Trigonometry equation problemShow that  arctan  ⁡  2  =  arccos  ⁡  (  1      /        5    )  .This is the result of solving one trigonometrical equation.

Rebecca Benitez · Accepted Answer

Draw the triangle. Opposite = 2. Adjacent = 1. Then use the Pythagorean theorem to find the hypotenuse, and recall that cosine = adjacent over hypotenuse.

Howard Nelson · Accepted Answer

Let   θ  =  arctan  ⁡  2. Then   tan  ⁡  θ  =  2 and we need to prove that   tan  ⁡  θ  =  tan  ⁡  (  arccos  ⁡  (  1      /        5    )  )  =  2. One way is to use the identity  tan  ⁡  x  =  ±            1                        cos                      2                          ⁡        x              −    1    ,which can be derived from the fundamental identity      cos          2        ⁡  x  +      sin          2        ⁡  x  =  1.For   x  =  1      /        5   we have that                    tan        ⁡        (        arccos        ⁡        (        1                  /                          5                )        )                            =                                          1                                          cos                                  2                                            ⁡              (              arccos              ⁡              (              1                              /                                            5                            )              )                                −          1                                                              (The sign is positive because         0                            &amp;lt;                    arccos        ⁡        (        1                  /                          5                )        &amp;lt;        π                  /                2        )                                                                    =                                          1                          (              1                              /                                            5                                            )                                  2                                                              −          1                =        2.

Trigonometry equation problem Show that arctan2 = arccos(1/sqrt5). This is the result of solving one trigonometrical equation.

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