You are asked to solve the equation [tex]8 cos^{2} x-3cosx=0 [/tex]. I have chosen to use x instead of theta but the work is the same.
Notice that you have a quadratic equation set equal to 0. The expression at left can be factored. We do this as follows:
[tex]cosx(8cosx-3)=0[/tex]
You have a product (at left) that equals to zero so one or both of the terms equals zero. That is,
cosx=0 and/or [tex]8cosx-3=0[/tex]
To solve the first we are looking for an angle measure between 0 and 180 degrees door which the cos of that angle is 0. You can use your calculator and input [tex]cos ^{-1}0= [/tex] and will find the answer to be 90 degrees. You might also know this from the graph of the cosine function.
The second equation can be solved as follows:
[tex]8cosx-3=0[/tex]
[tex]cosx=3/8[/tex]
You are looking for an angle whose cosine is 3/8 and can use the cosine inverse function on your calculator. Input [tex] Cos^{-1}(3/8)= [/tex]. The answer will be extremely close to 68 degrees and is 68 degrees when rounded.
Thus the two solutions we are looking for are 90 degrees and 68 degrees.
Please note that for this to work your calculator needs to be in degrees (not radians) and that you can control this in the “mode”.
