Question 282667: Solve the equation 5cos(2a)+3=0 for all values of a on the interval 0
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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
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'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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