SOLUTION: solve the following equation for theta, 0 < theta < 360, rounded to the nereast tenth of a degree. 2cos theta=5+6/cos theta 1.{27.6�} 2.{27.6�, 332.4�} 3.{152.4�, 207.6�}

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Question 1026502: solve the following equation for theta, 0 < theta < 360, rounded to the nereast tenth of a degree.
2cos theta=5+6/cos theta

1.{27.6�}
2.{27.6�, 332.4�}
3.{152.4�, 207.6�}
4. 2 and 3
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
answer is selection 3.

here's why.

start with 2 * cos(theta) = 5 + (6 / cos(theta)

multiply both sides of the equation by cos(theta) to get:

2 * cos^2(theta) = 5 * cos(theta) + 6

subtract the right side of the equation from both sides of the equation to get:

2 * cos^2(theta) - 5 * cos(theta) - 6 = 0.

this is a quadratic equation.

solve it using the quadratic formula.

you will get:

cos(theta) = -.8860009 or cos(theta) = 3.3860009

since cos(theta) has to be less than 1 or greater than -1, cos(theta) = 3.3860009 is not valid.

the only solution is cos(theta) = -.8860009.

the cosine function is negative in quadrants 2 and 3.

it is positive in quadrants 1 and 4.

to find the equivalent angle in quadrant 1, make the cosine positive.

you will get cos(theta) = .8860009.

solve for theta to get theta = arccos(.8860009) = 27.6250548 degrees.

the equivalent angle in the second quadrant is 180 degrees minus that.

the equivalent angle in the third quadrant is 180 degrees plus that.

you will get:

theta = 152.3749452 in the second quadrant and theta = 207.6250548 in the third quadrant.

round to nearest tenth of a degrees and you get:

theta = 152.4 in the second quadrant and theta = 207.6 in the third quadrant.

that agrees with selection 3.

find cos(152.4) with your calculator and you will see that it is equal to -.886.....

find cos(207.6) with your calculator and you will see that it is equal to -.886... as well.

the calculator confirms that the angles are correct.