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# How to find the set of angles when given the value of cos x?

Question:

Given: cos x = - 1/4, in which x is from 0 to 2pi. Find the set of the angle x.

Solution:

Because cos x = - 1/4 = - 0.25 < 0, so, the angle x is in Quadrant II or Quadrant IV.

Let x_{1} is an angle in Quadrant II. The terminal side of the angle x_{1} lies in Quadrant II. P is a point in the terminal side of the angle x_{1}. P has
coordinate (x, y). By definition, cos x_{1} = x/r, in which r = square root of x square plus y square, so r is always > 0. In Quadrant II, x is a negative number. So, cos x_{1} is
negative. So, there is an angle x_{1} will satisfy cos x = - 1/4.

Let x_{2} be an angle in Quadrant III. The terminal side of the angle x_{2} lies in Quadrant III. P is a point on the terminal side of the angle x_{2}. P has the
coordinate (x, y). By definition, cos x_{2} = x/r. Because in Quadrant III, x is a negative number, so, cos x_{2} is negative. So, there is an angle in Quadrant III that will
satisfy cos x = - 1/4.

Now we find the acute angle that corresponding to the absolute value of the given trigonometry function. That is, let cos x = |-1/4| = 1/4 = 0.25, then x = arc cos 0.25 = 75.52^{o}

The angle in Quadrant II is: x_{1} = 180^{o} - 75.52^{o} = 104.48^{o}

The angle in Quadrant III is: x_{2} = 180^{o} + 75.52^{o} = 255.52^{o}

Therefore, the set of the angles that satisfy cos x = - 1/4 and range from 0 to 2pi is {104.48^{o}, 255.52^{o}}

Look the figure above, the blue curve is the graph of y = cos x, the orange line is y = - 1/4. The intersection points of the blue curve ang orange line is x_{1} and x_{2}. Angle
x_{1} lies in Quadrant II and angle x_{2} lies in Quadrant III. x_{1} and x_{2} are the set of angles that satisfy cos x = - 1/4 and the angle x range from 0 to 2pi.
The angle set is {104.48^{o}, 255.52^{o}}. Watch the video for more details.