document.write( "Question 1170882: Consider the following.\r
\n" ); document.write( "\n" ); document.write( "t=-2pi/3\r
\n" ); document.write( "\n" ); document.write( "(a)Find the reference number t for the value of t.\r
\n" ); document.write( "\n" ); document.write( "t=?\r
\n" ); document.write( "\n" ); document.write( "(b)Find the terminal point determined by t.\r
\n" ); document.write( "\n" ); document.write( "(x,y)=(?,?)
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Algebra.Com's Answer #795777 by Theo(13342)\"\" \"About 
You can put this solution on YOUR website!
i believe the t stands for theta.
\n" ); document.write( "this is the usual designation for the angle.\r
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\n" ); document.write( "\n" ); document.write( "your equation is t = -2pi/3
\n" ); document.write( "that means the angle is equal to -2pi/3.
\n" ); document.write( "that would be in radians.
\n" ); document.write( "convert to degrees to get -2pi/3 * 180/pi = -2/3 * 180 = -2 * 60 = -120 degrees.
\n" ); document.write( "find the equivalent positive angle between 0 and 360 degrees by adding 360 until that occurs.
\n" ); document.write( "-120 + 360 = 240 degrees.
\n" ); document.write( "that's an angle in the third quadrant.
\n" ); document.write( "the reference angle to that will be the equivalent angle in the first quadrant.
\n" ); document.write( "that angle will be equal to 240 - 180 = 60 degrees.
\n" ); document.write( "if the angles are equivalent, then their trig function values will be the same except for the sign.
\n" ); document.write( "we'll do 60 degrees, 140 degrees, and -120 degrees.
\n" ); document.write( "we'll to sine, cosine, tangent.
\n" ); document.write( "cosecant, secant, and cotangent will not be done since they are reciprocals of since, cosine, tangent, and have the same sign as those.\r
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document.write( "angle            sine               cosine           tangent\r\n" );
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document.write( "60               .8660254038        .5               1.732050808\r\n" );
document.write( "240             -.8660254038       -.5               1.732050808\r\n" );
document.write( "-120            -.8660254038       -.5               1.732050808\r\n" );
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\r
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\n" ); document.write( "\n" ); document.write( "as you can see, the function values are the same except for the sign.
\n" ); document.write( "that makes the angles equivalent.\r
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\n" ); document.write( "\n" ); document.write( "note that:
\n" ); document.write( ".8660254038 is the same as sqrt(3)/3.
\n" ); document.write( ".5 is the same as 1/2.
\n" ); document.write( "1.732050808 is the same as sqrt(3).\r
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\n" ); document.write( "\n" ); document.write( "these are the trig functions of the common angle of 60 degrees which you may or may not remember.\r
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\n" ); document.write( "\n" ); document.write( "here's a table of the trig functions of the common angles.\r
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\n" ); document.write( "\n" ); document.write( "https://www.varsitytutors.com/hotmath/hotmath_help/topics/trigonometric-values-for-common-angles\r
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\n" ); document.write( "\n" ); document.write( "you could have done all this in radians without converting to degrees.
\n" ); document.write( "if so, you would have done the following:\r
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\n" ); document.write( "\n" ); document.write( "the angle is -2pi/3.
\n" ); document.write( "that'a a negative angle.
\n" ); document.write( "to convert to a positive angle between 0 and 2pi, add 2pi until you get there.
\n" ); document.write( "-2pi/3 + 2pi = -2pi/3 + 6pi/3 = 4pi/3.
\n" ); document.write( "the equivalent angle in degrees is 4pi/3 * 180/pi = 4*60 = 240, same as we got before when working in degrees.
\n" ); document.write( "the reference angle is 4pi/3 - p = 4pi/3 - 3pi/3 = pi/3.
\n" ); document.write( "the equivalent angle in degrees is pi/3 * 180/pi = 180/3 = 60 degrees, same as we got before when working in degrees.\r
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\n" ); document.write( "\n" ); document.write( "there is no one reference point for an angle of 240 degrees, unless you are talking about the unit circle.
\n" ); document.write( "if the point is on the unit circle, then the hypotenuse of the triangle formed would be equal to 1.
\n" ); document.write( "sine(240) = opposite / hypotenuse = opposite / 1 = -.8660254038.
\n" ); document.write( "that would be the y-coordinate.
\n" ); document.write( "cosine(240) = adjacent / hypotenuse = adjacent / 1 = -.5
\n" ); document.write( "that would be the x-coordinate.
\n" ); document.write( "the terminal point would therefore be (x,y) = (-.5,-.8660254038).\r
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\n" ); document.write( "\n" ); document.write( "on a graph, this would look like this.\r
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\n" ); document.write( "\n" ); document.write( "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.
\n" ); document.write( "all those points would be in the third quadrant.
\n" ); document.write( "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.\r
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\n" ); document.write( "\n" ); document.write( "note that the slope of the line is the tangent function.
\n" ); document.write( "that's because the slope is defined as the change in the value of y divided by the change in the value of x.
\n" ); document.write( "the change in the value of y was .8660254038.
\n" ); document.write( "the change in the value of x was .5.
\n" ); document.write( "that made the slope of the line shown equal to .8660254038 / .5 = 1.732050808.\r
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