The sum of all the values of x between 0 and 4 π which satisfy the equation s i n x 8 c o s 2 x = 1 is k π , then the value of k 5 is equal to

Question

The sum of all the values of x between 0 and 4 π which satisfy the equation s i n x 8 c o s 2 x = 1 is k π , then the value of k 5 is equal to,

MARKS App · Accepted Answer

Given equation is s i n x c o s 2 x = 1 2 2 ⇒ sin ⁡ x cos ⁡ x = 1 2 2 Case (I): When x > 0 i.e. x lies in 1 s t or 4 t h quadrant. sin ⁡ x cos ⁡ x = 1 2 2 ⇒ sin ⁡ 2 x = 1 2 ⇒ 2 x = π 4 , 3 π 4 , 9 π 4 , 11 π 4 , 17 π 4 , 19 π 4 ⇒ x = π 8 , 3 π 8 , 9 π 8 , 11 π 8 , 17 π 8 , 19 π 8 As x lies between 0 and 4 π , cos ⁡ x > 0 ⇒ x = π 8 , 3 π 8 , 17 π 8 , 19 π 8 Case (II): When cos ⁡ x 0 i.e., x lies in 2 n d or 3 r d quadrant. sin ⁡ 2 x = - 1 2 ⇒ 2 x = 5 π 4 , 7 π 4 , 13 π 4 , 15 π 4 , 21 π 4 , 23 π 4 ⇒ x = 5 π 8 , 7 π 8 , 13 π 8 , 15 π 8 , 21 π 8 , 23 π 8 As x lies between 0 and 4 π , cos ⁡ x 0 ⇒ x = 5 π 8 , 7 π 8 , 21 π 8 , 23 π 8 Sum of values of x = π 8 1 + 3 + 17 + 19 + 5 + 7 + 21 + 23 = 96 π 8 = 12 π ⇒ k = 12 ⇒ k 5 = 2 . 4

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