Question

URGENT!! Solve the following for θ, in radians, where 0≤θ<2π.

−4sin2(θ)−3sin(θ)+5=0
Select all that apply:

2.21
1.19
1.92
0.93
0.31
2.63

Answer 1

2.21

0.93

Explanation:

Given that;
`-4\sin^2\theta-3\sin \theta+5=0`

This is a quadratic equation is
`\sin \theta`, where
`a=-4,b=-3,c=5`

We want to solve for
`\theta` in radians, where 0≤θ<2π.

We apply the quadratic formula given by;

`\sin \theta=(-b\pm√(b^2-4ac) )/(2a)`

We substitute the given values to obtain;

`\sin \theta=(--3\pm√((-3)^2-4(-4)(5)))/(2(-4))`

Simplify;

`\sin \theta=(3\pm√(9+80))/(-8)`

`\sin \theta=(3\pm√(89))/(-8)`

`\sin \theta=0.804` or
`\sin \theta=-1.55`

When
`\sin \theta=0.804` ,
`\theta=\sin^(-1)(0.804)`

`\Rightarrow \theta=0.93` --In the first quadrant.

In the second quadrant;

`\theta=\pi-0.93=2.21`

When
`\sin \theta=-1.55` ,
`\theta` is not defined.

Answer 2

Answer: 0.93 radians & 2.21 radians

Explanation:

`-4sin^2\theta-3sin\theta+5=0\\\\\text{Since this is not factorable, use the quadratic formula to find the roots:}\\\\sin\theta=(-(-3)\pm √((-3)^2-4(-4)(5)))/(2(-4))\\\\\\.\quad=(3\pm √(9+80))/(-8)\\\\\\.\quad=(3\pm√(89))/(-8)\\\\\\.\quad=(3\pm9.43)/(-8)\\\\\\.\quad=(12.43)/(-8)\quad and\quad (-6.43)/(-8)\\\\\\.\quad=-1.55\quad and\quad 0.80\\\\\\\theta=sin^(-1)(-1.55)\quad and\quad \theta=sin^(-1)(0.80)`

`\theta=not\ valid\qquad and\quad \theta=0.927`

`\theta = 0.927\ radians\text{\ in the 1st quadrant and}\\\pi-0.927=2.21\ radians\text{\ in the 2nd quadrant}`