Question 1032307: I can't figure this type of trig question; evaluate inverse trig function when the number in the parenthesis is negative. I can get the answer when it is not negative inside the parenthesis. Help please!
Example: tan^-1 * (negative of the square root of three divided by three)
thanks.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if you make the function positive, then you will always get the angle in the first quadrant.
once you find the angle in the first quadrant, it's a relatively simple matter to find the quadrant where the function is negative and then find the equivalent angle in that quadrant.
looking at your example, i see arctan (minus the square root of 3) / 3).
the tangent function is negative in the second quadrant and the fourth quadrant.
it is positive in the first quadrant and the third quadrant.
solve for arctan (square root of 3) / 3.
your answer will be the angle in the first quadrant.
arctan (sqrt(3)/3 = 30 degrees.
the tangent is negative in the second quadrant and the fourth quadrant.
converting an angle from the first quadrant to any quadrant is given by the formulas.
second quadrant angle = 180 - first quadrant angle.
third quadrant angle = 180 + first quadrant angle.
fourth quadrant angle = 360 - first quadrant angle.
since your tangent function is negative, then you are looking for the equivalent angle in the second quadrant and the fourth quadrant.
your angle will be 180 - 30 = 150 in the second quadrant and it will be 360 - 30 = 330 in the fourth quadrant.
you can confirm using your calculator.
find tan(150) and you get -.5773502692.
find tan(330) and you get -.5773502692 as well.
use your calculator to find -sqrt(3)/3 and it will say that it is equal to -.5773502692.
this confirms the answer is -sqrt(3)/3.
your angles are either 150 degrees or 330 degrees.
both give you the tan function of -sqrt(3)/3 which is equal to -.5773502692.
in the first quadrant all the functions are positive.
in the second quadrant, sine is positive, cosine is negative, tangent is sine/cosine which is negative.
in the third quadrant, sine is negative, cosine is negative, tangent is sine/cosine which is positive.
in the fourth quadrant, sine is negative, cosine is positive, tangent is sine/cosine which is negative.
it's all based on the unit circle.
sine = y/hypotenuse
cosine = x/hypotenuse
tangent = y/x
in the first quadrant, y is positive and x is positive.
in the second quadrant, y is positive and x is negative.
in the third quadrant, y is negative and x is negative.
in the fourth quadrant y is negative and x is positive.
the hypotenuse is always positive.
if you graph y = tan(x), you can see the same solutions as well.
the graph is shown below:
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