Question 1170882: Consider the following.
t=-2pi/3
(a)Find the reference number t for the value of t.
t=?
(b)Find the terminal point determined by t.
(x,y)=(?,?)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i believe the t stands for theta.
this is the usual designation for the angle.
your equation is t = -2pi/3
that means the angle is equal to -2pi/3.
that would be in radians.
convert to degrees to get -2pi/3 * 180/pi = -2/3 * 180 = -2 * 60 = -120 degrees.
find the equivalent positive angle between 0 and 360 degrees by adding 360 until that occurs.
-120 + 360 = 240 degrees.
that's an angle in the third quadrant.
the reference angle to that will be the equivalent angle in the first quadrant.
that angle will be equal to 240 - 180 = 60 degrees.
if the angles are equivalent, then their trig function values will be the same except for the sign.
we'll do 60 degrees, 140 degrees, and -120 degrees.
we'll to sine, cosine, tangent.
cosecant, secant, and cotangent will not be done since they are reciprocals of since, cosine, tangent, and have the same sign as those.
angle sine cosine tangent
60 .8660254038 .5 1.732050808
240 -.8660254038 -.5 1.732050808
-120 -.8660254038 -.5 1.732050808
as you can see, the function values are the same except for the sign.
that makes the angles equivalent.
note that:
.8660254038 is the same as sqrt(3)/3.
.5 is the same as 1/2.
1.732050808 is the same as sqrt(3).
these are the trig functions of the common angle of 60 degrees which you may or may not remember.
here's a table of the trig functions of the common angles.
https://www.varsitytutors.com/hotmath/hotmath_help/topics/trigonometric-values-for-common-angles
you could have done all this in radians without converting to degrees.
if so, you would have done the following:
the angle is -2pi/3.
that'a a negative angle.
to convert to a positive angle between 0 and 2pi, add 2pi until you get there.
-2pi/3 + 2pi = -2pi/3 + 6pi/3 = 4pi/3.
the equivalent angle in degrees is 4pi/3 * 180/pi = 4*60 = 240, same as we got before when working in degrees.
the reference angle is 4pi/3 - p = 4pi/3 - 3pi/3 = pi/3.
the equivalent angle in degrees is pi/3 * 180/pi = 180/3 = 60 degrees, same as we got before when working in degrees.
there is no one reference point for an angle of 240 degrees, unless you are talking about the unit circle.
if the point is on the unit circle, then the hypotenuse of the triangle formed would be equal to 1.
sine(240) = opposite / hypotenuse = opposite / 1 = -.8660254038.
that would be the y-coordinate.
cosine(240) = adjacent / hypotenuse = adjacent / 1 = -.5
that would be the x-coordinate.
the terminal point would therefore be (x,y) = (-.5,-.8660254038).
on a graph, this would look like this.
any point on the line formed by the points (-.5,-.8660254038) and (0,0) can be viewed as a terminal point of the 240 degree angle.
all those points would be in the third quadrant.
it's only because we defined the terminal point as being on the unit circle, that we get the terminal point shown on the graph.
note that the slope of the line is the tangent function.
that's because the slope is defined as the change in the value of y divided by the change in the value of x.
the change in the value of y was .8660254038.
the change in the value of x was .5.
that made the slope of the line shown equal to .8660254038 / .5 = 1.732050808.
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