Question

find the value of sin suare24-sin square 6​

Answer 1

$\huge\bold\green{Solution♡}$

sin²24 - sin²6

= (sin24 - sin6)(sin24 + sin6)

= (2cos15 sin9 )(2sin15 cos9)

= (2sin15cos15)(2sin9cos9)

= (sin30)(sin18)

= (1/2)sin18.

sin72 = 2sin36cos36 = 4sin18cos18.(1 - 2 sin² 18).

But sin72 = cos18 so

1 = 4sin18(1 - 2 sin² 18) = 4x(1 - 2x²)

where x = sin18.

8x^3 - 4x + 1 = 0.

x = 1/2 is an obvious root, so remove a factor of 2x-1 :

8x³ - 4x + 1 = (4x²)(2x-1) + 4x² - 4x + 1

= (4x²)(2x - 1) + 2x(2x - 1) - (2x - 1)

= (4x² + 2x - 1) (2x - 1) = 0.

Either x = 1/2 or

x = (1/8)(-2 +/- √(4 - 4*4*(-1)))

= (2/8)(-1 + √5)

because sin18 > 0 so we must take the + sign.

Therefore :

sin²(24) - sin² (6) = (1/2)sin18 = (1/8)(√5 - 1).

$\huge\bold\purple{Hope \ It \ Helps ♡}$

Answer 2

Answer:

$\huge{\underline{\bold{\orange{Question:}}}}$

Find the value of sin suare24-sin square 6.

‌$\huge\fbox\red{Answer}$

Step-by-step explanation:

Solution♡

sin²24 - sin²6

= (sin24 - sin6)(sin24 + sin6)

= (2cos15 sin9 )(2sin15 cos9)

= (2sin15cos15)(2sin9cos9)

= (sin30)(sin18)

= (1/2)sin18.

sin72 = 2sin36cos36 = 4sin18cos18.(1 - 2 sin² 18).

But sin72 = cos18 so

1 = 4sin18(1 - 2 sin² 18) = 4x(1 - 2x²)

where x = sin18.

8x^3 - 4x + 1 = 0.

x = 1/2 is an obvious root, so remove a factor of 2x-1 :

8x³ - 4x + 1 = (4x²)(2x-1) + 4x² - 4x + 1

= (4x²)(2x - 1) + 2x(2x - 1) - (2x - 1)

= (4x² + 2x - 1) (2x - 1) = 0.

Either x = 1/2 or

x = (1/8)(-2 +/- √(4 - 4*4*(-1)))

= (2/8)(-1 + √5)

because sin18 > 0 so we must take the + sign.

Therefore :

sin²(24) - sin² (6) = (1/2)sin18 = (1/8)(√5 - 1).

Hope It Helps You ♡