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Question 1118764: solve for 0<=x<=360
sinx=cos^2x-1
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! start with sin(x) = cos^2(x) - 1
since sin^2(x) + cos^(x) = 1, then cos^2(x) = 1 - sin^2(x).
equation becomes sin(x) = 1 - sin^2(x) - 1
simplify to get sin(x) = -sin^2(x)
add sin^(x) to both sides of the equation to get sin^2(x) + sin(x) = 0
factor out sin(x) to get sin(x) * (sin(x) + 1) = 0
solve this quadratic equation to get sin(x) = 0 or sin(x) = -1
sin(x) = 0 when x = 0 degrees or x = 180 degrees or x = 360 degrees.
sin(x) = -1 when x = 270 degrees.
sin(x) = 1 when x = 0, 380, 360 degrees.
sin(x) is negative when x = 270 degrees.
those are your solutions.
you could have found this as follows:
when sin(x) = 0, x = 0 degrees found by use of your calculator.
the equivalent angle in the second quadrant is 180 - 0 = 180 degrees.
the equivalent angle in the third quadrant is 180 + 0 = 180 degrees.
the equivalent angle in the fourth quadrant is 360 - 0 = 360 degrees.
that gets you the possible angles of 0, 180, and 360.
you can use your calculator to confirm that sin(x) = 0 at all of those angles.
using your calculator, you will find that sin(x) = -1 leads to x = -90 degrees.
just put arcsin(-1) in your calculator and it will tell you that.
make sure your calculator is in degree mode before doing it.
the equivalent positive angle for -90 degrees is found by adding 360 to it.
you will get x = 270 degrees.
this angle could be in the third quadrant or it would be in the fourth quadrant.
it is on the border line between those quadrants.
if in the fourth quadrant, the equivalent angle in the first quadrant is 360 - 270 = 90 degrees.
if in the third quadrant, the equivalent angle in the first quadrant is 270 - 180 = 90 degrees.
either way, the equivalent angle in the first quadrant is 90 degrees.
the equivalent angle in the second quadrant is 180 - 90 = 90.
the equivalent angle in the third quadrant is 180 + 90 = 270.
the equivalent angle in the fouerth quadrant is 360 - 90 = 270.
either way, the possible angles in the first, second, third, or fourth quadrants is either 90 degrees or 170 degrees.
equivalent angle means that the trigonometric function gives you the same value except possibly for the sign.
sin(90) = 1
sin(270) = -1
you are looking for the angle between 0 and 360 that has a trig function of -1.
that has to be 270 degrees.
the angles between 0 and 360 that have their sine function = 0 or -1 would then be:
0, 180, 270, 360.
that's the interval of 0 <= x <= 360, where x is the angle.
that's your solution.
you can confirm that graphically by graphing y = xin(x) and y = cos^2(x) - 1.
the intersection of those 2 equations is your solution.
the graph looks like this:
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