8sin^2theta+10 sin theta cos theta =3cos^2 theta=0 then general solution for theta is
Assuming you meant
8sin^2θ+10 sinθ cosθ+3cos^2θ = 0
(2sinθ+cosθ)(4sinθ+3cosθ) = 0
see what you can do with that.
To solve the given equation, we can rearrange it to a quadratic form by using the identity: sin^2(theta) + cos^2(theta) = 1.
Given equation: 8sin^2(theta) + 10sin(theta)cos(theta) = 3cos^2(theta)
Rearranging using the identity:
8sin^2(theta) + 10sin(theta)cos(theta) - 3cos^2(theta) = 0
Now, let's substitute sin(theta) and cos(theta) with a single variable, say x.
Let x = sin(theta)
Using the identity sin^2(theta) + cos^2(theta) = 1, we get:
sin^2(theta) = 1 - cos^2(theta)
Substituting x back in, we have:
x^2 = 1 - cos^2(theta)
cos^2(theta) = 1 - x^2
Now, let's express cos(theta) using x:
cos(theta) = ±sqrt(1 - x^2)
Substituting the values back into the equation:
8x^2 + 10x(sqrt(1 - x^2)) - 3(1 - x^2) = 0
Simplifying the equation, we have:
8x^2 + 10x(sqrt(1 - x^2)) - 3 + 3x^2 = 0
11x^2 + 10x(sqrt(1 - x^2)) - 3 = 0
This is a quadratic equation in terms of x. To solve it, we can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
For our equation, the coefficients are:
a = 11
b = 10(sqrt(1 - x^2))
c = -3
Substituting these values into the quadratic formula, we get two possible values for x:
x = (-10(sqrt(1 - x^2)) ± sqrt((10(sqrt(1 - x^2)))^2 - 4(11)(-3))) / (2(11))
Simplifying it further, we have:
x = (-10(sqrt(1 - x^2)) ± sqrt(100(1 - x^2) + 132)) / 22
Now, we need to find the values of x that satisfy the initial substitution. Recall that x = sin(theta).
So, the possible values for sin(theta) can be found by substituting the values of x obtained from the quadratic equation back into sin(theta):
sin(theta) = x
Therefore, the general solution for theta can be found by taking the inverse sine of the possible values of sin(theta):
theta = arcsin(x)
These calculations will give you the general solutions for theta based on the given equation.