SOLUTION: Solve the equation 5cos(2a)+3=0 for all values of a on the interval 0<a<360. Round all answers to the nearest tenth of a degree.

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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