SOLUTION: Find the endpoint of the radius of the unit circle that makes a
−1590° angle with the positive horizontal axis.
Algebra ->
Trigonometry-basics
-> SOLUTION: Find the endpoint of the radius of the unit circle that makes a
−1590° angle with the positive horizontal axis.
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round that up to 5 and multiply that by 360 and you get 1800.
add 1800 degrees to -1590 degrees and you get 210 degrees.
every 360 degrees, your terminal point is the same on the unit circle.
210 degrees on the unit circle has the same terminal point as minus 1590 degrees on the unit circle.
since 210 degrees is between 0 and 360, that's the positive angle that you want to stop at.
that's the number of positive degrees from the positive x-axis.
if you want to know the number of negative degrees from the positive x-axis, then subtract 360 from that and you get -150 degrees.
subtract 360 from that 4 more times and you get -1590 degrees.
-1590 degrees is equivalent to negative 150 degrees from the positive x-axis and is also equivalent to positive 210 degrees from the positive x-axis.
negative degrees on the unit circle go around in a clockwise direction from the positive x-axis.
positive degrees on the unit circle go around in a counter-clockwise direction from the positive x-axis.
negative 1590 degrees has gone around 4.41666666667 times in a clockwise direction.
each time around is 360 degrees.
it looks like this on the unit circle.
the angles are equivalent because their trigonometric functions are the same.
sin(210) = -.5
sin(-1590) = -.5
sin(-150) = -.5
if you subtract 5 * 360 from 210, you get -1590.
if you graph sin(x) where x goes from -1590 to 210 degrees, the graph will repeat sin(x) = -.5 every 360 degrees and will look like this:
the reference angle is equal to 210 minus 180 = 30 degrees.
that's the angle that's within the triangle formed by 210 degrees.
it's also the angle that's within the triangle formed by -1590 degrees.
the reference angle looks like this:
the reference angle is 30 degrees which is 30 degrees from the negative x-axis.
it is the same for all of the angles shown because all of the angles shown are equivalent to each other since they all have the same value for their trigonometric functions.
the sine of 30 degrees in the third quadrant is negative because the side opposite the angle is negative and the hypotenuse is positive and sine equals opposite / hypotenuse, therefore negative.
right triangle ABC and ADE are similar triangles and therefore have the same corresponding angles.
the 30 degree angle is the same angle in both those triangles.
the 210 degree angle forms these right triangle in relation to the negative x-axis.
