Question 1008893
problem is:


2sin^2(theta) = 3 - 4cos(theta).


since sin^2(theta) = 1 - cos^2(theta), substitute for sin^2(theta) to get:


2 * (1 - cos^2(theta) = 3 - 4cos(theta)


simplify to get:


2 - 2cos^2(theta) = 3 - 4cos(theta)


add 2cos^2(theta) and subtract 2 from both sides of the equation to get:


0 = 3 - 4cos(theta) + 2cos^2(theta) - 2


combine like terms and flip the equation to get:


2cos^2(theta) - 4cos(theta) + 1 = 0


that's your quadratic equation that you need to factor.


use the quadratic formula to factor it as follows:


since the equation is in standard form of ax^2 + bx + c = 0, you get:


a = 2
b = -4
c = 1


the quadratic formula is:


<pre>
cos(theta) = -b plus or minus sqrt(b^2 - 4ac)
             --------------------------------
                            2a
</pre>


replace a,b,c with their values and you get:


<pre>
cos(theta) = -(-4) plus or minus sqrt((-4)^2 - 4*2*1)
             --------------------------------
                            2*2
</pre>


you will get:


cos(theta) = (4 + sqrt(8))/4
or:
cos(theta) = (4 - sqrt(8))/4


the result will be:


cos(theta) = 1.707106781
or:
cos(theta) = .2928932188


cosine can't be greater than 1, so the solution is:


cos(theta) = .2928932188


solve for theta to get:


theta = 72.96875154 degrees.


that would be in quadrant 1.


cosine is positive in quadrants 1 and 4 only.


your angle is therefore in quadrant 1 and in quadrant 4.


the angle in quadrant 4 is 360 - 72.96875154 = 287.0312485 degrees.


your solution is that theta = 72.96875154 or 287.0312485.


round that to a tenth of a degree and your solution is:


theta = 73.0 or 287.0 degrees.


the following picture shows the solution graphically.


2 equations were graphed.


they are:


y = sin^2(theta)


y = 3 - 4cos(theta)


the intersection of the 2 equations on the graph is when their values are equal.


that occurs at theta = 73 and theta = 287 between 0 and 360 degrees.


<img src = "http://theo.x10hosting.com/2015/120801.jpg" alt="$$$" </>


the graphical solution conforming to the algebraic solution is shown below:


this graph is of the equation y = 2cos^2(theta) - 4cos(theta) + 1


the solution, in this case, is when the graph crosses the x-axis.


<img src = "http://theo.x10hosting.com/2015/120802.jpg" alt="$$$" </>