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Question 1139919: If angle θ is a principal angle that lies in quadrant 3 such that 0° ≤ θ ≤360°, determine the exact values of x, y and r for tan θ = 2/3
Found 2 solutions by Alan3354, Theo:
Answer by Alan3354(69443)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! tan(theta) = 2/3
tan(theta) = opposite / adjacent.
this means that opposite is equal to 2 and adjacent is equal to 3.
since y is opposite the angle and x is adjacent the angle, then you get y = 2 and x = 3 in the first quadrant.
r is the hypotenuse of the right triangle formed.
the formula for r is r = sqrt(x^2 + y^2).
this makes r = sqrt(3^2 + 2^2) = sqrt(13).
in the third quadrant, y = -2 and x = -3.
tan(theta) = -2 / -3 = 2/3, same as in the first quadrant.
the value of the tangent function is the same whether or not the angle is in the first quadrant or in the third quadrant.
you can use your calculator to find theta.
arctan(2/3) = 33.69006753 degrees.
that's in the first quadrant.
that's also called the reference angle.
in the third quadrant, the equivalent angle would be 180 + 33.69006753 = 213.69006753 degrees.
the reference angle for 213.69006753 degrees is equal to 213.6906753 degrees minus 180 degrees = 33.6006753.
use your calculator to find tan(33.69006753) = .666666... = 2/3
use your calculator to find tan(213.69006753) = .66666.... = 2/3.
that means that 33.69006753 degrees and 213.69006753 degrees are equivalent angle because they have the same trigonometric values except for the signs.
sine(33.69006753) = .5547001962
sine(213.69006753) = -.5547001962
same value for sine except for the sign.
cosine(33.6906753)) = .8320502943
cosine(213.69006753) = -.8320502943
same value for cosine except for the sign.
we already did the tangent function.
secant is reciprocal of cosine, so the values will be the same except for the sign.
cosecant is reciprocal of sine, so the values will be the same except for the sign.
cotangent is reciprocal of tangent, so the values will be the same.
here's my picture of the angle in the first quadrant and the third quadrant.
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