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Question 1105064: Let tan(x)=2/5
What is the value of tan(2π−x)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! convert this to degrees and you can see it easier.
then convert to radians to get the same answer.
2*pi * 180/pi = 360 degrees.
if tan(x) = 2/5, then x = 21.80140949.
360 - 21.80140949 makes the equivalent angle in the fourth quadrant equal to 338.1985905.
tan(338.1985905) = -.4 which is the same as -2/5.
it's the same as tan(21.80140949) except the sign is negative.
if you did the same problem in radians, then you would get the following.
if tan(x) = 2/5, then x = .3805063771 radians.
tan(2*pi - x) = tan(6.283185307 - .3805063771) = tan(5.90267893 radians) = -.4 which is equal to -2/5.
you'll get the same answer, only you're dealing in radians rather than degrees.
.3805063771 radians * 180 / pi = 21.80140949 degrees.
5.90267893 radians * 180 / pi = 338.1985905 degrees.
the value of the tangent function for those angles is the same whether those angles are in degrees or radians.
the equivalent angle in the first quadrant is called the reference angle.
to find the reference angle from any quadrant, you use the following rules.
reference angle for an angle in the first quadrant is the angle itself.
no adjustment is necessary.
reference angle for an angle in the second quadrant is 180 minus the angle if you are in degrees and pi - the angle is you are in radians.
reference angle for an angle in the third quadrant is 180 plus the angle if you are in degrees and pi plus the angle if you are in radians.
reference angle for an angle in the fourth quadrant is 360 minus the angle if you are in degrees and 2*pi minus the angle if you are in radians.
180 degrees is equivalent to pi radians.
360 degrees is equivalent to 2*pi radians.
equivalent angles that are in different quadrants have the same value for their trigonometric function except for the sign.
for example:
assuming x is in the first quadrant, then tan(x) is positive.
the equivalent angle in the second quadrant is tan(180 - x) and it's tangent function is negative.
the equivalent angle in the third quadrant is tan(180 + x) and it's tangent function is positive.
the equivalent angle in the fourth quadrant is tan(360 - x) and it's tangent function is negative.
here's a picture representing the unit circle that might help, using the angle of 30 degrees.
in the unit circle, the hypotenuse is always equal to 1.
the length of the horizontal side of the triangle is the adjacent side to the angle.
the length of the vertical side of the triangle is the opposite side to the angle.
these change signs according to the quadrant they're in.
the hypotenuse is always positive.
the angle in the first quadrant is 30 degrees.
the angle in the second quadrant is 180 - 30 = 150 degrees.
the angle in the third quadrant is 180 + 30 = 210 degrees.
the angle in the fourth quadrant is 360 - 30 = 330 degrees.
tangent is side opposite / side adjacent = y/x.
in the first quadrant, tangent is equal to y/x = (1/2) / (sqrt(3)/2) = plus sqrt(3)/3.
in the second quadrant, the x value is negative, so tan(150) = opposite / adjacent = 1/2 / -sqrt(3)/2) = minus sqrt(3)/3.
in the third quadrant, the x and y values are negative, so tan(210) = opposite / adjacent = -1/2 / -sqrt(3)/2 = plus sqrt(3)/3.
in the fourth quadrant, the x is positive and the y is negative, so tan(330) = opposite / adjacent = 1/2 / -sqrt(3)/2 = minus sqrt(3)/3.
the tangent is positive in quadrants 1 and 3.
the tangent is negative in quadrants 2 and 4.
the angles are equivalent so they have the same tangent function but with different signs depending on the quadrant they're in.
the reference angle for the angles of 30, 150, 210, and 330 are all 30 degrees which is the equivalent angle in the first quadrant.
here's your picture.
the angle in the picture is 30 degrees.
your angle is 21.880140949 degrees or .3805063771 radians.
you could draw the same circle using your angle and the measurements of your opposite and adjacent sides, only in that case, it would not be a unit circle because the hypotenuse would be sqrt(29) rather than 1.
the same concept applies, however, regardless of the length of the hypotenuse.
here's a quick picture of what your circle would look like.
the circle is not drawn.
only the triangle formed by the angle in each quadrant is shown.
the hypotenuse of each of the triangles is the radius of the circle that would be formed if it were shown.
tangent = y/x.
vertical side is y.
horizontal side is x.
in quadrant 1, it's plus y / plus x = plus 2 / plus 5 = plus 2/5.
in quadrant 2, it's plus y / minus x = plus 2 / minus 5 = minus 2/5.
in quadrant 3, it's minus y / minus x = minus 2 / minus 5 = plus 2/5.
in quadrant 4, it's minus y / plus x = minus 2 / plus 5 = minus 2/5.
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