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