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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 x1 is an angle in Quadrant II. The terminal side of the angle x1 lies in Quadrant II. P is a point in the terminal side of the angle x1. P has coordinate (x, y). By definition, cos x1 = 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 x1 is negative. So, there is an angle x1 will satisfy cos x = - 1/4.

Let x2 be an angle in Quadrant III. The terminal side of the angle x2 lies in Quadrant III. P is a point on the terminal side of the angle x2. P has the coordinate (x, y). By definition, cos x2 = x/r. Because in Quadrant III, x is a negative number, so, cos x2 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 = 75.52o

The angle in Quadrant II is: x1 = 180o - 75.52o = 104.48o

The angle in Quadrant III is: x2 = 180o + 75.52o = 255.52o

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

how to find the set of the angles when given the value of the trigonometry function.

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 x1 and x2. Angle x1 lies in Quadrant II and angle x2 lies in Quadrant III. x1 and x2 are the set of angles that satisfy cos x = - 1/4 and the angle x range from 0 to 2pi. The angle set is {104.48o, 255.52o}. Watch the video for more details.