Question

The smallest positive angle x for which the equation, 8 sinx.cos 5x-8 sin 5x.cos x=1 holds is equal to ______degrees.

Answer 1

Given:

8 sinx.cos 5x-8 sin 5x.cos x=1

To find:

The smallest positive angle x for which the equation, 8 sinx.cos 5x-8 sin 5x.cos x=1 holds is equal to ______degrees.

Solution:

From given, we have,

8 sinx.cos ^5x - 8 sin ^5x.cos x = 1

taking sinx.cosx common, we get,

8 sinx.cosx (cos ^4x - sin ^4x) = 1

we use the formula,

a² - b² = (a + b) (a - b)

⇒ cos ^4x - sin ^4x = (cos ^2x + sin ^2x) (cos ^2x - sin ^2x)

so we get,

8 sinx.cosx  (cos ^2x + sin ^2x) (cos ^2x - sin ^2x) = 1

we  know that, (cos ^2x + sin ^2x) = 1

8 sinx.cosx  (1) (cos ^2x - sin ^2x) = 1

8 sinx.cosx  (1) (cos ^2x - sin ^2x) = 1

4 × 2 sinx.cosx  (1) (cos ^2x - sin ^2x) = 1

we use the formula, sin 2a = 2 sina cosa

and we use the formula cos ^2a - sin ^2a = cos 2a

4 sin 2x (cos 2x) = 1

2 × 2 sin 2x (cos 2x) = 1

we use the formula, sin 2a = 2 sina cosa

2 sin 4x = 1

sin 4x = 1/2 = sin π/6

4x = π/6

x = π/24 = 180/24

∴ x = 7.5

Therefore, the smallest positive angle x is 7.5°

Answer 2

Answer:

X=7.5*

Step-by-step explanation:

Given:

8 sin x. cos 5x-8 sin 5x.cos x=1

To find:

The smallest positive angle x for which the equation, 8 sin x. cos 5x-8 sin 5x.cos x=1 holds is equal to ______degrees.

Solution:

From given, we have,

8 sin x. cos ^5x - 8 sin ^5x.cos x = 1

taking sin x. cos x common, we get,

8 sin x. cos x (cos ^4x - sin ^4x) = 1

we use the formula,

a² - b² = (a + b) (a - b)

⇒ cos ^4x - sin ^4x = (cos ^2x + sin ^2x) (cos ^2x - sin ^2x)

so we get,

8 sin x. cos x  (cos ^2x + sin ^2x) (cos ^2x - sin ^2x) = 1

we  know that, (cos ^2x + sin ^2x) = 1

8 sin x. cos x  (1) (cos ^2x - sin ^2x) = 1

8 sin x. cos x  (1) (cos ^2x - sin ^2x) = 1

4 × 2 sin x. cos x  (1) (cos ^2x - sin ^2x) = 1

we use the formula, sin 2a = 2 sin a cos a

and we use the formula cos ^2a - sin ^2a = cos 2a

4 sin 2x (cos 2x) = 1

2 × 2 sin 2x (cos 2x) = 1

we use the formula, sin 2a = 2 sin a cos a

2 sin 4x = 1

sin 4x = 1/2 = sin π/6

4x = π/6

x = π/24 = 180/24

∴ x = 7.5

Therefore, the smallest positive angle x is 7.5°