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Find all real numbers in the interval [ 0 , 2pi ) that satisfy each equation. Round approximate answers to the nearest tenth.


1. sin 2x - sin x cos x = cos x


2. sin 2x cos x - cos 2x sin x = -1/2


3. cos^2(theta / 2) = sec (theta)


4. sin (2x) = 3 sin(x)



 Apr 30, 2019

1) \(sin(2x)-sin(x)cos(x)=cos(x)\)

For the sin (2x), you can use the Double Angle Formula of \(sin(2x)=2sin(x)cos(x)\).


Then the \(2sin(x)cos(x)-sin(x)cos(x)\) simplifies to \(sin(x)cos(x)\).


Then subtract cos(x) from both sides.


You can factor the cos(x) out.


Then, set both terms equal to 0.


Add 1 to both sides on the sin(x) function.


Then, take the inverse of both functions.


The only values that satisfy those are \(x=90^o, x=270^o\) or \(x=\frac{\pi}{2},x=\frac{3\pi}{2}\).

 Apr 30, 2019

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