What is the exact value of 1 tan ⁡ 20 ∘ − 1 sin ⁡ 80 ∘ ?By putting the above into a calculator, I get 3 , but I cannot seem to be able to do it algebraically.

Question

What is the exact value of       1          tan      ⁡              20                  ∘                      −      1          sin      ⁡              80                  ∘                    ?By putting the above into a calculator, I get       3  , but I cannot seem to be able to do it algebraically.

Melany Flynn · Accepted Answer

Colored for easier reading.                               1                      tan            ⁡                          20                              ∘                                                    −                  1                      sin            ⁡                          80                              ∘                                                                        =                                          cos            ⁡                          20                              ∘                                                          sin            ⁡                          20                              ∘                                                    −                  1                      cos            ⁡                          10                              ∘                                                                                      =                                                        cos              ⁡                              20                                  ∘                                                                    2              sin              ⁡                              10                                  ∘                                            cos              ⁡                              10                                  ∘                                                              −                      1                          cos              ⁡                              10                                  ∘                                                                                                      =                                          cos            ⁡                          20                              ∘                                      −            2            sin            ⁡                          10                              ∘                                                          2            sin            ⁡                          10                              ∘                                      cos            ⁡                          10                              ∘                                                                                      =                              2                                    cos              ⁡                              60                                  ∘                                            cos              ⁡                              20                                  ∘                                            −              sin              ⁡                              10                                  ∘                                                                    sin              ⁡                              20                                  ∘                                                                                                      =                                          cos            ⁡                          80                              ∘                                      +            cos            ⁡                          40                              ∘                                      −            2            sin            ⁡                          10                              ∘                                                          sin            ⁡                          20                              ∘                                                                                      =                                                        cos              ⁡                              40                                  ∘                                            −              cos              ⁡                              80                                  ∘                                                                    sin              ⁡                              20                                  ∘                                                                                                      =                                          cos            ⁡                          40                              ∘                                      +            cos            ⁡                          100                              ∘                                                          sin            ⁡                          20                              ∘                                                                                      =                                                        2              cos              ⁡                              70                                  ∘                                            cos              ⁡                              30                                  ∘                                                                    sin              ⁡                              20                                  ∘                                                                                                      =                    2        cos        ⁡                  30                      ∘                          =                  3

Mark Elliott · Accepted Answer

In any case for computing the result you can use the following trigonometric identities:  tan  ⁡  (      80          ∘        )  =  tan  ⁡  (      π    3    +      20          ∘        )  =            tan      ⁡      (              π        3            )      +      tan      ⁡      (              20                  ∘                    )              1      −      tan      ⁡      (              π        3            )      tan      ⁡      (              20                  ∘                    )        =                    3            +      tan      ⁡      (              20                  ∘                    )              1      −              3            tan      ⁡      (              20                  ∘                    )      Now, let&#039;s target   tan  ⁡  (      20          ∘        ) to find also the value for   tan  ⁡  (      80          ∘        ). We can write:  tan  ⁡  (      60          ∘        )  =  tan  ⁡  (      20          ∘        +      20          ∘        +      20          ∘        )  =            3      tan      ⁡      (              20                  ∘                    )      −              tan        3            ⁡      (              20                  ∘                    )              1      −      3              tan        2            ⁡      (              20                  ∘                    )      Setting   x  =  tan  ⁡  (      20          ∘        ) in the last equation we obtain:       3    =            3      x      −              x        3                    1      −      3              x        2            . And hence (assuming   1  −  3  x  ≠  0):      x    3    −  3      3        x    2    −  3  x  +      3    =  0I think you can go from here.

What is the exact value of 1/(tan20^∘)−1/(sin80^∘) ? By putting the above into a calculator, I get sqrt3, but I cannot seem to be able to do it algebraically.

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