Question 907641: find the value of the expression:
sinx = -1/2, 180 degrees < x <270 degrees find tanx
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! sin(x) = -1/2
that puts x either in the third or fourth quadrant since sine is negative in the third and fourth quadrant while sine is positive in the first and second quadrant.
in the first quadrant, sin(x) would be equal to 1/2 since all trig functions are positive in the first quadrant.
if sine(x) is equal to 1/2, then x is equal to 30 degrees because sin(30) = 1/2.
your triangle is a 30/60/90 triangle which has sides that are in the ratio of:
sin(30) = 1/2 = opposite / hypotenuse.
this makes opposite side = 1 and hypotenuse = 2
by pythagorus, opposite side squared plus adjacent side squared = hypotenuse squared.
solve for adjacent side to get adjacent side to 30 degree angle is equal to sqrt(3).
tan(30) is therefore equal to side opposite divided by side adjacent which is equal to 1/sqrt(3) and 1/sqrt(3) is equivalent to 3/sqrt(3) after you multiply it by sqrt(3)/sqrt(3) to simplify it by rationalizing the denominator.
this makes:
sin(30) = 1/2
cos(30) = sqrt(3)/2
tan(30) = 1/sqrt(3) = sqrt(3)/3
since your interval is between 180 degrees and 270 degrees, then the angle has to be in the third quadrant.
in the third quadrant, the angle is 180 + 30 = 210 degrees.
in the third quadrant, the tangent will be positive because tangent is equal to sine divided by cosine and sine and cosine are both negative in the third quadrant.
a negative divided by a negative is equal to a positive, making the tangent positive in the third quadrant.
your solution is that the tangent of 210 degrees is equal to sqrt(3)/3.
you can use your calculator to confirm this is true.
using your calculator, tan(210) = .5773502692
using your calculator, sqrt(3)/3 = .5773502692
they're equivalent.
a picture of your triangles formed and the angles formed is shown below.
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