How do you graph #theta=-840^circ#?

Question

Anna Andrews · Accepted Answer

graph{-tan(2pi/3)x [-10, 10, -5, 5]}

First you simplify the angle so it's between 0 and 360º #-840º = -360º - 480º = -360º - 360º - 120º# So we have the equation #theta = -120º# That means it's 120 degrees clockwise from the #x^+# semiaxis (because of the sign), or in another way, 240 degrees counter-clockwise from the #x^+# semiaxis. Since we only have an angle, the radius can be any real value, so our equation describes a line. Just find the appropriate angle on the graph paper and trace a line through it. Or, if you only have rectangular graph paper / no protactor nearby, calculate the rate of growth of the line (#tan(-120)#) and take two points within that line to trace it.

Quinn Anderson · Answer

undefined To graph $ \theta = -840^\circ $, you would first convert the angle to its equivalent angle within one full revolution. Since $ 360^\circ $ represents one full revolution, adding or subtracting multiples of $ 360^\circ $ does not change the position of the angle on the graph.

To find the equivalent angle within one revolution, you would add or subtract multiples of $ 360^\circ $ until you obtain an angle between $ 0^\circ $ and $ 360^\circ $.

In this case, to find the equivalent angle:

$$ -840^\circ + 360^\circ = -480^\circ $$
$$ -480^\circ + 360^\circ = -120^\circ $$

So, $ \theta = -840^\circ $ is equivalent to $ \theta = -120^\circ $ within one full revolution.

Now, to graph $ \theta = -120^\circ $, you would locate the angle $ -120^\circ $ on the unit circle or on a graph with polar coordinates. Since $ -120^\circ $ is measured clockwise from the positive x-axis, you would start from the positive x-axis and rotate clockwise by $ 120^\circ $. This places the point on the unit circle at $ (0.5, -\sqrt{3}/2) $.

Therefore, to graph $ \theta = -840^\circ $, you would graph the point $ (0.5, -\sqrt{3}/2) $.