I want to calculate cos ⁡ π 12 . Using the sum-of-angles formula and using the half-angle formula results in different answers. cos ⁡ π 12 = cos ⁡ π / 6 2 = 1 2 + cos ⁡ π 6 2 = 1 2 + 3 4 cos ⁡ π 12 = cos ⁡ π 3 cos ⁡ π 4 + sin ⁡ π 3 sin ⁡ π 4 = ( 1 2 ) ( 2 2 ) + ( 3 2 ) ( 2 2 ) = 2 + 6 4 Why does this happen? Shouldn't these be the same?

Question

I want to calculate   cos  ⁡      π    12  . Using the sum-of-angles formula and using the half-angle formula results in different answers.  cos  ⁡      π    12    =  cos  ⁡            π              /            6        2    =            1      2        +                  cos        ⁡                  π          6                    2        =            1      2        +                  3            4        cos  ⁡      π    12    =  cos  ⁡      π    3    cos  ⁡      π    4    +  sin  ⁡      π    3    sin  ⁡      π    4    =      (          1      2        )        (                  2            2        )    +      (                  3            2        )        (                  2            2        )    =                    2            +              6              4  Why does this happen? Shouldn&#039;t these be the same?

dobradisamgn · Accepted Answer

Those may not look the same, but they are equal! Notice the first one is wrapped completely in a square root, so rewrite the second one so that it too is wrapped in a single square root:                    2            +              6              4    =                    (                                            2                        +                          6                                4                )            2        =                    2        +        2                  2                          6                +        6            16        =                    8        +        4                  3                    16        =            1      2        +                  3            4

Why do the sum-of-angles and half-angle formulas return different answers for cos (pi)/(12)?

Answered question

Answer & Explanation

New Questions in Trigonometry