How can the trigonometric equation be proven? 2 sin ⁡ ( 80 ° ) = sin ⁡ ( 60 ° ) sin ⁡ ( 100 ° ) × sin ⁡ ( 50 ° ) sin ⁡ ( 20 ° )

Question

How can the trigonometric equation be proven?  2  sin  ⁡  (  80      °    )  =            sin      ⁡      (      60              °            )              sin      ⁡      (      100              °            )        ×            sin      ⁡      (      50              °            )              sin      ⁡      (      20              °            )

postotnojeyf · Accepted Answer

sin            ⁡                          80              ∘                        sin            ⁡                          20              ∘                        sin            ⁡                          100              ∘                                            sin            ⁡                          50              ∘                                                          =                              sin            ⁡                          80              ∘                        sin            ⁡                          20              ∘                        ⋅            2            sin            ⁡                          50              ∘                        cos            ⁡                          50              ∘                                            sin            ⁡                          50              ∘                                                                        =        2        sin        ⁡                  80          ∘                sin        ⁡                  20          ∘                cos        ⁡                  50          ∘                                                  =        2                  (          −                      1            2                    (          cos          ⁡                      100            ∘                    −          cos          ⁡                      60            ∘                    )          )                cos        ⁡                  50          ∘                                                  =        (        cos        ⁡                  60          ∘                −        cos        ⁡                  100          ∘                )        cos        ⁡                  50          ∘                                                  =                  1          2                cos        ⁡                  50          ∘                −        cos        ⁡                  100          ∘                cos        ⁡                  50          ∘                                                  =                  1          2                cos        ⁡                  50          ∘                −                  1          2                (        cos        ⁡                  150          ∘                +        cos        ⁡                  50          ∘                )                                          =        −                  1          2                ×                  (          −                                    3                        2                    )                                                  =                              3                    4                                                  =                  1          2                sin        ⁡                  60          ∘

drzwiczkih5a · Accepted Answer

2  ⋅  sin  ⁡      80    ∘    ⋅                              sin          ⁡                      100            ∘                          ⏟                    sin      ⁡              80        ∘              ⋅  sin  ⁡      20    ∘    =  2  ⋅  sin  ⁡      80    ∘    ⋅  2  sin  ⁡      40    ∘                                cos          ⁡                      40            ∘                          ⏟                    sin      ⁡              50        ∘              ⋅  sin  ⁡      20    ∘    =    =                              4          ⋅          sin          ⁡                      80            ∘                    ⋅          sin          ⁡                      40            ∘                    ⋅          sin          ⁡                      20            ∘                          ⏟                            3                    /            2        ⋅  sin  ⁡      50    ∘    =  sin  ⁡      60    ∘    ⋅  sin  ⁡      50    ∘    .

How can the trigonometric equation be proven? 2 sin(80°)=(sin(60°))/(sin(100°)) xx (sin(50°))/(sin(20°))

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