Question 1119527: Solve the equation in the interval [0,\, 2 \pi]. If there is more than one solution write them separated by commas.
Hint: To solve this problem you will need the quadratic formula, inverse trigonometric functions and the symmetry of the unit circle.
(\cos x)^2 - 1.3 \cos x - 1.98 =0
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the problem is:
cos^2(x) - 1.3 * cos(x) - 1.98 = 0
let y = cos(x).
the problem becomes:
y^2 - 1.3 * y - 1.98 = 0
use the quadratic formula to get:
y = -.9 or y = 2.2
since y = cos(x), you get:
cos(x) = -.9 or cos(x) = 2.2
working this problem in radians, and solving for the angle in the first quadrant, where all trig functions are positive, you get:
arccos(.9) = .4510268118 or arccos(2.2) = undefined, since cosine function has a maximum value of 1.
your solution in the first quadrant is that the angle is .4510268118 radians.
your cosine function is negative, however,.
that means the angle can only be in the second quadrant or the third quadrant.
the equivalent angle in the second quadrant is x = pi - .4510268118 = 2.690565842 radians.
the equivalent angle in the third quadrant is x = pi + .4510268118 = 3.592619465 radians.
if you graph the function of cos^2(x) - 1.3 * cos(x) - 1.98, you will see that the value of the function is 0 when x = 2.690565842 and when x = 3.592619465.
round these values to 3 decimal places, and they become x = 2.691 radians and x = 3.593 radians as shown on the graph.
here's the graph.
the unshaded area of this graph goes from x = 0 to x = 2 * pi.
that's the interval for the value of x that you want.
the maximum value of cos(x) is plus or minus 1, and the maximum value of cos^2(x) is also plus or minus 1.
note that 2 * pi is equal to 2 * 3.141592654 which is equal to 6.283185307.
the unshaded area of the graph therefore goes from x = 0 to x = approximately 6.283, when rounding to 3 decimal places.
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