Why doesn't sin ⁡ a + sin ⁡ 3 a + sin ⁡ 5 a cos ⁡ a + cos ⁡ 3 a + cos ⁡ 5 a = tan ⁡ 3 a imply sin ⁡ a + sin ⁡ 5 a = 0 and cos ⁡ a + cos ⁡ 5 a = 0?

Question

Why doesn&#039;t             sin      ⁡      a      +      sin      ⁡      3      a      +      sin      ⁡      5      a              cos      ⁡      a      +      cos      ⁡      3      a      +      cos      ⁡      5      a        =  tan  ⁡  3  a imply   sin  ⁡  a  +  sin  ⁡  5  a  =  0 and   cos  ⁡  a  +  cos  ⁡  5  a  =  0?

mainzollbtt · Accepted Answer

Let&#039;s look at a simple example to see why it isn&#039;t true.Let   α  =            π      2      , then                    sin        ⁡        α        +        sin        ⁡        5        α                            =        sin        ⁡                              π            2                          +        sin        ⁡                                            5              π                        2                                                            =        1        +        1                                          =        2        ≠        0.

Owen Mathis · Accepted Answer

Consider the following true equality:            5      +      6      +      2              3      +      6      +      4        =  1But       6    6    =  1 as well. Does this imply that the remaining terms 5,2,3,4 must be zero? Obviously not. You cannot simply eliminate 6 from the numerator and the denominator 6, nor can you conclude that the remaining terms are zero. This doesn&#039;t follow any of the rules of arithmetic.

Why doesn't (sin a+sin 3a+sin 5a)/(cos a+cos 3a+cos 5a)=tan 3a imply sin a+sin 5a=0 and cos a+cos 5a=0?

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