To simplify the equation, we can use the trigonometric identities:
Using these identities, we can rewrite the equation as:
6sin^2(x) - 11cos(x) - 10 = 0
Let y = sin(x)
6y^2 - 11√(1 - y^2) - 10 = 0
Let z = 1 - y^2
6(1 - z) - 11√z - 10 = 0
6 - 6z - 11√z - 10 = 0
-6z - 11√z - 4 = 0
Let u = √z
-6u^2 -11u - 4 = 0
Solving this quadratic equation for u, we get:
u = (-(-11) ± √((-11)^2 - 4(-6)(-4))) / (2*(-6))u = (11 ± √(121 - 96)) / -12u = (11 ± √25) / -12
u = (11 ± 5) / -12
u1 = (11 + 5) / -12 = 16 / -12 = -4/3u2 = (11 - 5) / -12 = 6 / -12 = -1/2
Since z = u^2:
z1 = (-4/3)^2 = 16/9z2 = (-1/2)^2 = 1/4
Since z = 1 - y^2:
1 - y^2 = 16/9 (or) 1 - y^2 = 1/4
y^2 = 1 - 16/9 (or) y^2 = 1 - 1/4
y^2 = 9/9 - 16/9 (or) y^2 = 4/4 - 1/4
y^2 = -7/9 (or) y^2 = 3/4
Since y^2 cannot be negative, the equation y^2 = -7/9 has no solution.
y = ± √(3/4)y = ± √(3)/2
Therefore, the solutions to the equation are:
y = √(3)/2 and y = -√(3)/2
Since y = sin(x):
sin(x) = √(3)/2, sin(x) = -√(3)/2
The solutions for x are:
x = π/3 + 2πn, x = 2π/3 + 2πnWhere n is an integer.
To simplify the equation, we can use the trigonometric identities:
cos(π/2 - θ) = sin(θ)sin(3π/2 + θ) = -cos(θ)Using these identities, we can rewrite the equation as:
6sin^2(x) - 11cos(x) - 10 = 0
Let y = sin(x)
6y^2 - 11√(1 - y^2) - 10 = 0
Let z = 1 - y^2
6(1 - z) - 11√z - 10 = 0
6 - 6z - 11√z - 10 = 0
-6z - 11√z - 4 = 0
Let u = √z
-6u^2 -11u - 4 = 0
Solving this quadratic equation for u, we get:
u = (-(-11) ± √((-11)^2 - 4(-6)(-4))) / (2*(-6))
u = (11 ± √(121 - 96)) / -12
u = (11 ± √25) / -12
u = (11 ± 5) / -12
u1 = (11 + 5) / -12 = 16 / -12 = -4/3
u2 = (11 - 5) / -12 = 6 / -12 = -1/2
Since z = u^2:
z1 = (-4/3)^2 = 16/9
z2 = (-1/2)^2 = 1/4
Since z = 1 - y^2:
1 - y^2 = 16/9 (or) 1 - y^2 = 1/4
y^2 = 1 - 16/9 (or) y^2 = 1 - 1/4
y^2 = 9/9 - 16/9 (or) y^2 = 4/4 - 1/4
y^2 = -7/9 (or) y^2 = 3/4
Since y^2 cannot be negative, the equation y^2 = -7/9 has no solution.
y = ± √(3/4)
y = ± √(3)/2
Therefore, the solutions to the equation are:
y = √(3)/2 and y = -√(3)/2
Since y = sin(x):
sin(x) = √(3)/2, sin(x) = -√(3)/2
The solutions for x are:
x = π/3 + 2πn, x = 2π/3 + 2πn
Where n is an integer.