Use the fact that the terminal point associated with moving a distance t = π / 6units anti-clockwise around the unit circle, x 2 + y 2 = 1, has coordinates ( x , y ) = ( 3 / 2 , 1 / 2 ),and the method from Example 63 in the lectures to determine the exact coordinates of theterminal point associated with moving a distance t = p i / 12 units ant-clockwise around theunit circle. Hence determine the exact values of sin ⁡ ( π / 12 ) , cos ⁡ ( π / 12 ) and tan ⁡ ( π / 12 ) .

Question

Use the fact that the terminal point associated with moving a distance   t  =  π      /    6units anti-clockwise around the unit circle,       x    2    +      y    2    =  1, has coordinates   (  x  ,  y  )  =  (      3        /    2  ,  1      /    2  ),and the method from Example 63 in the lectures to determine the exact coordinates of theterminal point associated with moving a distance   t  =  p  i      /    12 units ant-clockwise around theunit circle. Hence determine the exact values of   sin  ⁡  (  π      /    12  )  ,  cos  ⁡  (  π      /    12  )     and     tan  ⁡  (  π      /    12  )  .

cegukwt · Accepted Answer

Step 1  cos  ⁡  (  π      /    6  )  =      6        /    2  ,     sin  ⁡  (  π      /    6  )  =  1      /    2  sin  ⁡  (  π      /    12  )  =                    1        −        cos        ⁡        (        π                  /                6        )            2        ⇒  sin  ⁡  (  π      /    12  )  =                    1        −        (                  3                          /                2        )            2        ⇒  sin  ⁡  (  π      /    12  )  =                    (        2        −                  3                )                  /                2            2        ⇒  sin  ⁡  (  π      /    12  )  =                    2        −                  3                    4        ⇒  sin  ⁡  (  π      /    12  )  =                    2        −                  3                    2        cos  ⁡  (  π      /    12  )  =                    1        +        cos        ⁡        (        π                  /                6        )            2        ⇒  cos  ⁡  (  π      /    12  )  =                    1        +        (                  3                          /                2        )            2        ⇒  cos  ⁡  (  π      /    12  )  =                    (        2        +                  3                )                  /                2            2        ⇒  cos  ⁡  (  π      /    12  )  =                    2        +                  3                    4        ⇒  cos  ⁡  (  π      /    12  )  =                    2        +                  3                    2      Step 2  tan  ⁡  (  π      /    12  )  =            sin      ⁡      (      π              /            12      )              cos      ⁡      (      π              /            12      )        ⇒  tan  ⁡  (  π      /    12  )  =                    2        −                  3                    2                      2        +                  3                    2        ⇒  tan  ⁡  (  π      /    12  )  =            2      −              3                    2      +              3              ⇒  tan  ⁡  (  π      /    12  )  =                    2        −                  3                            2        +                  3                      ⇒  tan  ⁡  (  π      /    12  )  =                    (        2        −                  3                )        (        2        −                  3                )                    (        2        +                  3                )        (        2        −                  3                )              ⇒  tan  ⁡  (  π      /    12  )  =                    (        2        −                  3                          )          2                            (                  2          2                −                              3                    2                )              ⇒  tan  ⁡  (  π      /    12  )  =                    (        2        −                  3                          )          2                            4        −        3              ⇒  tan  ⁡  (  π      /    12  )  =                    (        2        −                  3                          )          2                    1        ⇒  tan  ⁡  (  π      /    12  )  =  2  −      3  coordinates of the terminal point associated with   t  =  π      /    12 are   (            2      +              3              2    ,            2      −              3              2    )

Use the fact that the terminal point associated with moving a distance t = pi/6 units anti-clockwise around the unit circle, x^2+y^2 = 1, has coordinates (x,y) = (sqrt3/2, 1/2).

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