Find All Solutions of the Equation Tanx 1

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Question 214062: Find all solutions of tan x = 1
Answer by Theo(11806) (Show Source):
You can put this solution on YOUR website!
tan x = 1
tan 45 degrees = 1 *****
add 90 degrees to this and you get 135 degrees
tan 135 degrees = -1
add 90 degrees to this and you get 225 degrees
tan 225 degrees = 1 *****
add 90 degrees to this and you get 315 degrees
tan 315 degrees = -1
add 90 degrees to this and you get 405 degrees
now:
405 degrees - 360 degrees = 45 degrees and you're back where you started.
every 360 degrees you will be back where you started.
135 degrees + 360 = 495 degrees.
this is the same place on the graph.
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your question was:
find all solutions of tan(x) = 1
looking at the above, we see that tan(x) = 1 when:
x = 45 degrees
x = (180 + 45) = 225 degrees
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If you subtract 180 from 45 degrees you will get:
-135 degrees.
tan -135 degrees = 1
this is not to be unexpected because -135 degrees is equivalent to 225 degrees.
just add 360 to it and you get:
-135 + 360 = 225 degrees.
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whether you add 180 to 45 degrees and keep adding 180 to the result you will get either the equivalent of 45 degrees or 225 degrees.
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whether you subtract 180 from 45 degrees and keep subtracting 180 from the result you will get either the equivalent of 45 degrees or 225 degrees.
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why is tan (x) equal to 1 at 45 and 225 degrees and equal to -1 at the others?
it has to do with the sign of the result.
tan (x) = opposite divided by hypotenuse = y/x on the graph because the angle being measured is between the x axis and the line from the origin to the point (x,y).
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on the graph, x is the horizontal direction and y is the vertical direction.
in the first quadrant (45 degrees), x and y are both positive. divide a + by a + and you get a +.
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in the second quadrant (135 degrees), x is negative and y is positive. divide a - by a + and you get a -.
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in the third quadrant (225 degrees), x is negative and y is negative. divide a - by a - and you get a +.
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in the fourth quadrant (315 degrees), x is positive and y is negative. divide a + by a - and you get a -.
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i have some pictures that will show you what is happening.
go to the following website and click on problem number 214062. There may be more than one picture up there. You will see them as 214062.1, 214062.2, etc.
click on each in turn if that is the case. if the pictures aren't there when you look, wait 30 minutes and try again. they will be there.
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the website to go to is:
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http://theo.x10hosting.com/

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your answer is:
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45 + (180*n) for all real integer values of n
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you could probably get away with all integer values of n since I suppose they are all real. I can't think of an instance where they wouldn't.
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here's how that would work:
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45 + (180*0) = 45 which is good.
45 + (180*1) = 45 + 180 = 225 which is good.
45 + (180*2) = 45 + 360 = 405 which is equivalent to 45 so that is good.
45 + (180*3) = 45 + 540 = 585 which is equivalent to 225 so that is good.
45 - (180*1) = 45 - 180 = -135 which is equivalent to 225 so that is good.
45 - (180*2) = 45 - 360 = -315 which is equivalent to 45 so that is good.
etc.
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pictures to follow:
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