<PRE><B>Question 282667</B>
Equation is:


5*cos(2a)+3=0


Subtract 3 from both sides of this equation to get:


5*cos(2a) = -3


Divide both sides of this equation by 5 to get:


cos(2a) = -3/5 = -.6 


arccos(-3/5) = 126.8698976 degrees.


This means that 2a = 126.8698976 degrees.


This means that a = 63.43494882 degrees.


I thought this might be the only one, but it turns out I was wrong.


There is another angle between 0 and 360 whose cosine is equal to -.6.


I found that out through graphing, and then confirmed through the use of the calculator.


It turns out the other angle is 233.1301024 degrees.


I should have known based on the following facts.


The cosine is negative in quadrants 2 and 3 only.


Within each quadrant, the value of the cosine either goes from 1 to 0 or 0 to -1.


This means that the angle I was looking for had to be in quadrants 2 and 3 which means that there had to be 2 angles I was looking for, and not 1.


The graph of the cosine of x confirms that.


That graph is shown below


{{{graph (400,200,-7,7,-1,1,cos(x),-.6)}}}


The graph is in radian format.


0 to 360 degrees goes from 0 radians to 6.283 radians.


This corresponds to x = 0 to x = 6.283.


The resolution isn&#39;t that great, so I would use 6.3 as a rough measure.


You can see that the graph of cosine (x) goes from 1 at 0 radians to 0 at 1.6 radians to -1 at 3.1 radians to 0 at 4.7 radians to 1 at 6.3 radians.


The value of x is the value of the radians.


The line at y = -.6 intersects the graph of the equation of cosine(x) at 2 points between 0 and 6.3 radians.


Those intersections are at:


x = 2.2 radians and at:


x = 4.1 radians.


This corresponds to the angles at:


126.8698976 degrees, and at:


233.1301024 degrees.


Those angles, however, are double the angles you are looking for.


Those correspond to 2 times the angle you are looking for.


cos(2a) = -.6 is the equation.


Once you find 2a, you have to cut it in half to find a.


The angles you are looking for are therefore:


a = 63.43494882 and a = 116.5650512


Rounded to the nearest 10th of a degree, those angles becomes:


a = 63.4 and a = 116.6 degrees.


To confirm these angles were good, I substituted in the original equation to get:


5*cos(2a) = -3


This equation became:


5*cos(126.8698976) = -3 which became 5 * -.6 = -3 which became -3 = -3.


5*cos(233.1301024) = -3 which became 5 * -.6 = -3 which became -3 = -3.


 











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