Question 921665: Why is tan(-569) = -tan29
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! to understand why, you have to understand what reference angles are.
here's some references that might help you with that.
http://www.regentsprep.org/regents/math/algtrig/ATT3/referenceAngles.htm
http://www.regentsprep.org/regents/math/algtrig/ATT3/referenceTriangles.htm
http://academics.utep.edu/Portals/1788/CALCULUS%20MATERIAL/4_4%20TRIF%20FNS%20OF%20ANGLES.pdf
to make a long story short, the reference angle for -529 is equal to 29 degrees.
that is found as follows:
start with -569
add 360 to get -209
add 360 to get 151
an angle of 151 degrees occupies the same position in the unit circle that an angle of -569 degrees occupies.
they are effectively the same angles as far as the unit circle is concerned.
now that you are in the positive angle range of 0 to 360 degrees, you can find the reference angle.
the formula to find the reference angle is as follows:
if the angle is in quadrant 1, then the reference angle is the angle.
if the angle is in quadrant 2, then the reference angle is 180 minus the angle.
if the angle is in quadrant 3, then the reference angle is 180 plus the angle.
if the angle is in quadrant 4, then the reference angle is 360 minus the angle.
151 is in quadrant 2, so the reference angle is 180 - 151 = 29 degrees.
the reference angle and the angle will have the same value for the trigonometric function with the exception of the sign.
in quadrant 1, all trigonometric functions are positive.
in quadrant 2, sine is positive, cosine is negative, tangent is negative.
in quadrant 3, sine is negative, cosine is negative, tangent is positive.
in quadrant 4, sine is negative, cosine is positive, tangent is negative.
from the perspective of the tangent function, it is positive in quadrant 1 and negative in quadrant 2 and positive in quadrant 3 and negative in quadrant 4.
since the angle of 151 degrees is in quadrant 2, the tangent function is negative.
it has the same value as the reference angle.
the sign, however, is different.
this is why tangent of 151 is equal to minus the tangent of 29.
if you understand the unit circle, you will understand why the signs change.
let h represent hypotenuse
in the unit circle, y represents the side opposite the angle and x represents the side adjacent the angle.
you get:
sine = y/h
cosine = x/h
tangent = y/x
in quadrant 1, x and y are positive.
in quadrant 2, x is negative and y is positive.
in quadrant 3, x and y are negative.
in quadrant 4, x is positive and y is negative.
this is why the trig functions have different signs, depending on the quadrant they are in.
in quadrant 1, tangent = y/x = +/+ = +
in quadrant 2, tangent = y/x = +/- = -
so, you have tan(151) = -tan(29).
the following graph of the tangent function will show you that the angles have the same tangent value every 180 degrees.
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