Question 889997: If I have tanA= -2/6, what two possible locations could I show for it on the coordinate axis for the terminal arm of angle A, and how do I find two possible values for the measure of angle A and the related acute angle?
So far I have found the tan inverse of the radical (-2/6) for which I got -18.4, and I have plotted the point (-2,6) on quadrant II. Now I'm unsure how to continue properly to get the correct related acute angle, where exactly I can place another terminal arm for this angle (so far I've put one in quadrant I from 0 to point (2,6), though perhaps I shouldn't put one there since tan is positive in quad 1) as well as how to find another value of angle A.
Thank you so much for any way you can help!
-Hanna
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! tan is minus in quadrants 2 and 4.
tan is plus in quadrants 1 and 3.
you want tan to be minus so you want quadrants 2 and 4.
find the angle in quadrant 1.
to do that, make tan positive instead of negative.
the calculator will always find you the angle in quadrant 1 if you make the function value positive.
the angle will be arctan(2/6) = 18.4349...
that angle in quadrant 2 will be 180 - 18.4349... = 161.5650...
that angle in quadrant 4 will be 360 - 18.4349... = 341.5650...
your related acute angle is the angle in quadrant 1 that you already found.
that's called the reference angle.
there's another way to do it just using the numbers given.
you are given that tan(A) = -(2/6)
that makes tan negative.
you still have to know that tan is negative in quadrants 2 and 4.
since tan on the unit circle is equal to y/x, you get:
y = +/- 2
x = +/- 6
for tan to be negative, you have to have either y/x = -2/+6 or y/x = +2/-6
y/x = -2/+6 occurs in quadrant 4 because y is negative and x is positive in quadrant 4.
y/x = +2/-6 occurs in quadrant 2 because y is positive and x is negative in quadrant 2.
either way, you get the same angles in the same quadrants.
the following picture should show you what i mean.
see below the picture for further comments.
if you use your calculator to find tan(-.333...), the calculator will tell you that the angle is equal to -18.4332...
that's nice to know but not very helpful when you want the equivalent angle in quadrant 2.
it's easiest to find the equivalent angle in quadrant 1 and then work from there.
the reference angle for an angle in any quadrant is the equivalent angle in quadrant 1.
finding the angle for the positive value of the function is the quickest way to find the reference angle.
you had tan = -2/6.
find angle whose tan is equal to +2/6.
the calculator will tell you the value of that function in quadrant 1.
that's your reference angle.
from quadrant 1, you can then easily go to any other quadrant by using the formulas for those quadrants.
angle in quadrant 2 is equal to 180 - angle in quadrant 1.
angle in quadrant 3 is equal to 180 + angle in quadrant 1.
angle in quadrant 4 is eqjual to 360 - angle in quadrant 1.
regardless of the method that you chose, you still had to know that tan is minus in quadrants 2 and 4.
a quick review:
sine is positive in quadrants 1 and 2.
sine is negative in quadrants 3 and 4.
cosine is positive in quadrants 1 and 4.
cosine is negative in quadrants 2 and 3.
tangent is equal to sine / cosine.
that makes tangent positive in quadrants 1 and 3. that's +/+ in quadrant 1 and -/- in quadrant 3.
that makes tangent negative in quadrants 2 and 4. that's +/- in quadrant 2 and -/+ in quadrant 4.
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