Question 1203221
<font color=black size=3>
<font color=red>Answers:</font>
theta = 70.529 (approximate)
theta = 180 (exact)
theta = 289.471 (approximate)
Round the decimal values however your teacher instructs.



Explanation


I'll use x in place of theta (symbol {{{theta}}})
The reason for this is to allow us to graph on the xy grid.


The equation to solve is
{{{3cos^2(x)+2cos(x) = 1}}}
which can be rearranged to 
{{{3cos^2(x)+2cos(x) - 1 = 0}}}


Let
{{{f(x) = 3cos^2(x)+2cos(x) - 1}}}
The goal is to find the roots or x-intercepts.


Open up your favorite graphing app.
I'll go with Desmos. GeoGebra is another good option.
A calculator like a TI83 or TI84 works also.


We will need Desmos to be in degrees mode. Click the wrench icon in the upper right corner. Then swap from "radians" to "degrees".


In Desmos, we'll type in f(x) =  3cos^2(x)+2cos(x) - 1
To restrict the domain, append {0 <= x < 360} 


The full input is <font color=red>f(x) = 3cos^2(x)+2cos(x) - 1 {0 <= x < 360}</font>


Check out this link to see what I mean
<a href="https://www.desmos.com/calculator/dbjhvzx0cg">https://www.desmos.com/calculator/dbjhvzx0cg</a>
Refer to the first box in the left-hand panel.


The curve looks like a strange W of sorts.
It has three x-intercepts
x = 70.529 (approximate)
x = 180 (exact)
x = 289.471 (approximate)
Those are the 3 solutions to the equation.



Side notes:<ul><li>Unfortunately Desmos appears to only allow accuracy up to 3 decimal places. Use GeoGebra for better precision.</li><li>WolframAlpha is another good solver to use. There are plenty others.</li><li>The graph window in Desmos is: xmin = -50, xmax = 400, ymin = -6, ymax = 6</li><li>In Desmos you can click on the x intercept location to have its coordinates show up</li></ul>
</font>