Question

If the distance between the points (asin 42 ,0 )and (0, sin 78 a) is d,then d^2-a^2

Answer 1

Step-by-step explanation:

(asin42-0)²+(0-sin78)²=d²

solve this u get urs answer perfectly

Answer 2

The value of  d² - a² is - a²/2 cos 36°.

Given: The distance between the points ( a sin 42°, 0 ) and ( 0, a sin 78° ) is d.

To Find: The value of d² - a².
Solution:

• We know that the distance between two points can be found using the formula,

        Distance = √ ( ( x2 - x1 )² + ( y2 - y1 )² )                       ......(1)

• We will also need a formula for trigonometrical identities,

        sin²Ф = ( 1 - cos 2Ф ) / 2                                              .......(2)

        cos C + cos D = 2 cos ( C+D )/2 × cos ( C - D )/2        .......(3)

       

Using the distance formula with points ( a sin 42°, 0 ) and ( 0, a sin 78° ) from (1), we get;

         Distance = √ ( ( x2 - x1 )² + ( y2 - y1 )² )      

     ⇒  d² = a² sin² 42° + a² sin² 78°

     ⇒  d² = a² [ ( 1 - cos 84° )/2 + ( 1 - cos 156° )/2 ]                  [using (2) ]

     ⇒  d² = a²/2 [ 2 - ( cos 84° + cos 156° ) ]                             [ using (3) ]

     ⇒  d² = a²/2 [ 2 - ( 2×cos 60° × cos 36°) ]

     ⇒  d² = a² [ 1 - cos 60° × cos 36° ]

     ⇒  d² = a² [ 1 - 1/2 × cos 36° ]

     ⇒  d² = a² - a²/2 cos 36°

     ⇒  d² - a² = - ( a²/2 ) × cos 36°

Hence, the value of  d² - a² is - ( a²/2 ) × cos 36°.

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