Question 1116912
your equation is sec(x) + 4cos(x) = 5


sec(x) is equivalent to 1/cos(x).


your equation becomes 1/cos(x) + 4cos(x) = 5


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


1 + 4cos^2(x) = 5cos(x)


subtract 5cos(x) from both sides of the equation to get 1 + 4cos^2(x) - 5cos(x) = 0


reorder the terms in descending order of degree to get:


4cos^2(x) - 5cos(x) + 1 = 0


factor this quadratic equation to get:


cos(x) = .25 or cos(x) = 1


in the first quadrant, cos(x) = 1 when x = 0 degrees.


in the first quadrant, cos(x) = .25 when x = 75.52248781.


cos(x) is positive in the first and fourth quadrant.


in the fourth quadrant, 0 degrees is equal to 360 - 0 = 360 degrees.


in the fourth quadrant, 75.52248781 degrees is equal to 360 - 75.52248781 = 284.4775122 degrees.


your solution is that:


x = 0, 75.52248781, 284.4775122, 360 degrees in the interval from 0 to 360 degrees.


that would be the interval 0 <= x <= 360 degrees.


in the interval 0 < x < 360 degrees, then your only solutions are x = 75.52248781 or x = 284.4775122.


here's a graph of your equation with the possible solutions shown.


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