Why am I getting a different value for sin ⁡ ( 2 tan − 1 ⁡ 4 3 ) than my calculator?Way 1:If I punch the above expression in my calculator, I get 24 25 Way 2: sin ⁡ ( 2 tan − 1 ⁡ ( 4 3 ) ) sin ⁡ ( tan − 1 ⁡ ( 2 × 4 3 1 − ( 4 3 ) 2 ) ) [ Using the formula 2 arctan ⁡ ( x ) = arctan ⁡ ( 2 x 1 − x 2 ) ] sin ⁡ ( tan − 1 ⁡ ( − 24 7 ) ) − sin ⁡ ( tan − 1 ⁡ ( 24 7 ) ) − sin ⁡ ( sin − 1 ⁡ ( 24 25 ) ) − 24 25 Why am I getting a different answer than that of my calculator?

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Why am I getting a different value for   sin  ⁡      (    2          tan              −        1              ⁡          4      3        )   than my calculator?Way 1:If I punch the above expression in my calculator, I get       24    25  Way 2:  sin  ⁡      (    2          tan              −        1              ⁡          (              4        3            )        )    sin  ⁡      (          tan              −        1              ⁡          (                        2          ×                      4            3                                    1          −          (                      4            3                                )                          2                                          )        )    [      Using the formula           2      arctan      ⁡      (      x      )      =      arctan      ⁡              (                              2            x                                1            −                          x              2                                      )              ]  sin  ⁡      (          tan              −        1              ⁡          (                        −          24                7            )        )    −  sin  ⁡      (          tan              −        1              ⁡          (              24        7            )        )    −  sin  ⁡      (          sin              −        1              ⁡          (              24        25            )        )    −      24    25  Why am I getting a different answer than that of my calculator?

Tasinazzokbc · Accepted Answer

One has to keep in mind, which quadrant the angle lies in from the beginning, and not derive it from calculations. Your calculations will be prone to mistakes because trigonometric and inverse trigonometric functions are not one-to-one functions.Notice that       tan          −      1        ⁡  4      /    3  &amp;gt;      tan          −      1        ⁡  1  =  π      /    4. So,   θ  =  2      tan          −      1        ⁡  4      /    3  &amp;gt;  π      /    2 and belongs to second quadrant. However       tan          −      1        ⁡  (negative value) lies in fourth quadrant. So the conversion  sin  ⁡      (                  2                  tan                      −            1                          ⁡                  (                      4            3                    )                    II quadrant        )    →  sin  ⁡      (                            tan                      −            1                          ⁡                  (                                    −              24                        7                    )                    IV quadrant        )  is incorrect. But it can be corrected as follows  sin  ⁡      (                  2                  tan                      −            1                          ⁡                  (                      4            3                    )                    II quadrant        )    =  sin  ⁡      (                  π        +                  tan                      −            1                          ⁡                  (                                    −              24                        7                    )                    II quadrant        )  Now calculation should give correct answer.

Paula Cameron · Accepted Answer

Notice that your formula to simplify the arctan gives a positive answer on the left hand side  2  arctan  ⁡      4    3    &amp;gt;  0but a negative on the right  arctan  ⁡            8      3              1      −                        (                      4            3                    )                2              &amp;lt;  0The formula itself is the mistake.

Why am I getting a different value for sin(2 tan^(-1) 4/3) than my calculator?

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