Choose the point on the terminal side of -210°.

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Choose the point on the terminal side of -210°.

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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  • phi
-210 means start at the x-axis and go clockwise 210 degrees |dw:1441116254558:dw|
i know that but i dont know where to go from there
  • phi
if you want the (x,y) numbers, we need to know the length of the radius Probably they are assuming the radius is length 1

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Other answers:

  • phi
210 is 30 "past" 180, so we can draw this triangle |dw:1441116408316:dw|
how did you get one for the radius
is it just always assumed
  • phi
It's a 30-60-90 triangle, so (if you learned this) the opposite side (which is the "y" value) is 1/2 of the hypotenuse if the radius is 1 , then y= 0.5 x = \( \sqrt{3} \cdot \text{ opposite side} = \frac{\sqrt{3}}{2}\)
  • phi
we can't answer the question unless we know r. But usually we use the "unit circle" when they ask questions like this
so how would i get the answer from that
is it\[(-\sqrt{3}, 1)\]
  • phi
do you know the lengths of the sides of a 30-60-90 triangle if you know the hypotenuse is 1?
  • phi
more generally, do you know about sin, cos ?
yes i know about that but im just so confused how you got the 2 sides in the first place
i just tried to find the 3rd side too and i dont think its right
  • phi
one step at a time. Let's say we have a unit circle (which means r=1) and we pick a random spot on the circle |dw:1441117005906:dw|
  • phi
that is an example with 2 different spots notice that the (x,y) coordinates of a point are also the lengths of the sides (x is the horizontal side, and y is the vertical side) to make a triangle, we draw in a line from the point to the origin. |dw:1441117192063:dw|
  • phi
we have a right triangle. the hypotenuse will always be 1 (it is the radius) we will always measure the angle up (or down) from the x-axis |dw:1441117274941:dw|
  • phi
does this make sense so far?
yes it does thank you
  • phi
so let's say we have this triangle |dw:1441117338853:dw|
  • phi
do you know the sin 30º ?
which is 0.5?
you mean the coordinates?
  • phi
yes. and notice the opposite side is also the "height" of the point above the x-axis. this is the y value of the (x,y) coords
how do we find x then
  • phi
what is the definition of cosine ?
adj/hyp
  • phi
yes, and in this special case where we set the hyp=1 we have cos theta= adjacent
  • phi
notice the adjacent side is the distance along the x-axis
do i use the pythagorean theorem
  • phi
we could, but we could use cos 30 = x people generally memorize this value
what would that equal
  • phi
sqr(3)/2 They expect you to memorize sin, cos and tan of 0, 30, 45, 60, and 90 degrees so put that on your list of things to do.
  • phi
also, drill into your head the idea that cos = x sin = y (for a UNIT circle)
  • phi
now , I said cos 30 = sqr(3)/2 but we have to be careful. the x value of the point is on the left side , so it is negative. the point is at \[ \left( - \frac{\sqrt{3}}{2} , \frac{1}{2} \right)\]
  • phi
if we measure the angle "normally" from the positive x-axis |dw:1441117944457:dw| the angle is 150º and your calculator will give the answer cos 150= -0.866.. (which is decimal for - sqr(3)/2) and sin(150) = 0.5 (decimal for 1/2) but without a calculator, we figure out the reference angle 30 use our memorized value cos 30= sqr(3)/2 , then look at the picture and see we want the x value (the cos ) to be negative, so -sqr(3)/2 and sin 30 = 0.5, and because we are on the positive (above the x-axis) we know it is +0.5
  • phi
for more help, see https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func/Trig-unit-circle/v/unit-circle-definition-of-trig-functions-1

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