We have the expression 13 sin ⁡ [ tan − 1 ⁡ ( 12 5 ) ].Apparently the answer is 12, and I have to simplify it, and I'm assuming it means I have to show it's 12, without using a calculator.Normally I show my own work in the questions, but in this case I have absolutely no clue how to. The only thing I know that might help is that tan ⁡ ( x ) = sin ⁡ ( x ) cos ⁡ ( x )

Question

We have the expression   13  sin  ⁡  [      tan          −      1        ⁡  (            12      5        )  ].Apparently the answer is 12, and I have to simplify it, and I&#039;m assuming it means I have to show it&#039;s 12, without using a calculator.Normally I show my own work in the questions, but in this case I have absolutely no clue how to. The only thing I know that might help is that   tan  ⁡  (  x  )  =                    sin        ⁡        (        x        )                    cos        ⁡        (        x        )

Sanaa Hinton · Accepted Answer

There is a right triangle with side lengths 5, 12 and 13. Draw this triangle, and choose one of the two non-right angles t for which  tan  ⁡  t  =      12    5  Recall that the tangent is the opposite side over the adjacent side.

Ximena Skinner · Accepted Answer

Let             tan              −        1              ⁡          12      5        =    θ            ⟹        (    i    )    tan    ⁡    θ    =          12      5      and       (    i    i    )    −          π      2        ≤    θ    ≤          π      2                ⟹        cos    ⁡    θ    ≥    0            ⟹        cos    ⁡    θ    =          1              sec        ⁡        θ              =    +          1              1        +                  tan          2                ⁡        θ              =    ⋯            ⟹        sin    ⁡    θ    =    tan    ⁡    θ    ⋅    cos    ⁡    θ    =    ⋯

We have the expression 13 sin ⁡ [ tan −<!

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