+25 Re-arrange to get the equation equal to 0.
\(4cos^2(\theta) + 9sin(\theta) - 6 = 0\)
Solve like any other quadratic equation.
\(cos(\theta ) = \frac{-9\frac{+}{-}\sqrt{81 - 4(4*(-6))}}{8}\)
And you should be able to get it from there.
Solve for theta over the real numbers:
4 cos^2(theta) = 6-9 sin(theta)
Subtract 6-9 sin(theta) from both sides:
-6+4 cos^2(theta)+9 sin(theta) = 0
Write -6+4 cos^2(theta)+9 sin(theta) = 0 in terms of sin(theta) using the identity cos^2(theta) = 1-sin^2(theta):
-2+9 sin(theta)-4 sin^2(theta) = 0
The left hand side factors into a product with three terms:
-((sin(theta)-2) (4 sin(theta)-1)) = 0
Multiply both sides by -1:
(sin(theta)-2) (4 sin(theta)-1) = 0
Split into two equations:
sin(theta)-2 = 0 or 4 sin(theta)-1 = 0
Add 2 to both sides:
sin(theta) = 2 or 4 sin(theta)-1 = 0
sin(theta) = 2 has no solution since for all theta element R, -1<=sin(theta)<=1 and True:
4 sin(theta)-1 = 0
Add 1 to both sides:
4 sin(theta) = 1
Divide both sides by 4:
sin(theta) = 1/4
Take the inverse sine of both sides:
Answer: | theta = pi-sin^(-1)(1/4)+2 pi n_1 for n_1 element Z
or theta = sin^(-1)(1/4)+2 pi n_2 for n_2 element Z
+129845 4cos^2θ = 6 - 9sin θ
4(1 -sin^2 θ) = 6 - 9sinθ
4 - 4sin^2θ = 6 - 9sin θ re-arrange as
4sin^θ - 9sinθ + 2 = 0 factor as
(4sinθ - 1) (sin θ - 2) = 0
Set each factor to 0.........the second one will have no solution since sinθ = 2 is impossible
4sinθ = 1
sinθ = 1/4
And this happens at about 14.48° + n*360° and at about 165.52° + n360°.....where n is an integer
