What are the EIGHT values of `theta` that satisfy the following equation: `45sin^2(2theta)+13sin(2theta)-2=0` on `0<= theta < 360` ** **

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The equation `45*sin^2(2*x) + 13*sin(2x) - 2 = 0` has to be solved.

If `sin 2x = X`

=> `45*X^2 + 13*X - 2 = 0`

=> `X = -2/5` and `X = 1/9`

=> `sin 2x = -2/5` and `sin 2x = 1/9`

`2x = sin^-1(-2/5)`

=> `x = (sin^-1(-2/5))/2`

=> `x ~~ 101.78` , `x ~~ 168.21`

`2x = sin^-1(1/9)`

=> `x = (sin^-1(1/9))/2`

=> `x ~~ 3.18` and `x ~~ 86.8` 1

But the graph of `y = 45*sin^2(2*x) + 13*sin(2x) - 2` shows that there are 8 values of x in [0, 360] where `45*sin^2(2*x) + 13*sin(2x) - 2 = 0` . This is due to a reduction in periodicity of the the function sin(2*x) to 180 degrees from 360 degrees for sin x.

The other values of x that satisfy the equation are 183.18, 266.81, 269.99, 348.21

**The roots of the equation are: 101.78, 168.21, 3.18, 86.81, 183.18, 281.78, 266.81, 348.21 degrees.**

You should solve the trigonometric equation using the following substitution, such that:

`sin 2theta = t`

Changing the variable, yields:

`45t^2 + 13t - 2 = ` `0`

Using quadratic formula yields:

`t_(1,2) = (-13+-sqrt(169 + 360))/(90)`

`t_(1,2) = (-13+-sqrt529)/90`

`t_(1,2) = (-13+-23)/90 => t_1 = 1/9, t_2 = 2/5`

Solving back for `theta` yields:

`sin 2theta = 1/9 => 2theta = sin^(-1)(1/9) => theta_1 = 0.05 rad`

Since the sine function is positive in quadrant 2 yields:

`theta_2 = 3.14 - 0.05 => theta_2 = 3.09 rad`

`sin 2theta = 2/5 => 2theta = sin^(-1)(2/5) => theta_3 = 0.20 rad`

`theta_4 = 3.14 - 0.20 => theta_4 = 2.94 rad`

**Hence, evaluating the solutions to the given equation, in radians, under the given conditions, yields **`theta_1 = 0.05 rad, theta_2 = 3.09 rad, theta_3 = 0.20 rad, theta_4 = 2.94 rad.`

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