## anonymous one year ago Choose the point on the terminal side of -210°.

1. phi

-210 means start at the x-axis and go clockwise 210 degrees |dw:1441116254558:dw|

2. anonymous

i know that but i dont know where to go from there

3. 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

4. phi

210 is 30 "past" 180, so we can draw this triangle |dw:1441116408316:dw|

5. anonymous

how did you get one for the radius

6. anonymous

is it just always assumed

7. 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 = $$\sqrt{3} \cdot \text{ opposite side} = \frac{\sqrt{3}}{2}$$

8. phi

we can't answer the question unless we know r. But usually we use the "unit circle" when they ask questions like this

9. anonymous

so how would i get the answer from that

10. anonymous

is it$(-\sqrt{3}, 1)$

11. phi

do you know the lengths of the sides of a 30-60-90 triangle if you know the hypotenuse is 1?

12. phi

more generally, do you know about sin, cos ?

13. anonymous

yes i know about that but im just so confused how you got the 2 sides in the first place

14. anonymous

i just tried to find the 3rd side too and i dont think its right

15. 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|

16. 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|

17. 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|

18. phi

does this make sense so far?

19. anonymous

yes it does thank you

20. phi

so let's say we have this triangle |dw:1441117338853:dw|

21. phi

do you know the sin 30º ?

22. anonymous

which is 0.5?

23. anonymous

you mean the coordinates?

24. 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

25. anonymous

how do we find x then

26. phi

what is the definition of cosine ?

27. anonymous

28. phi

yes, and in this special case where we set the hyp=1 we have cos theta= adjacent

29. phi

notice the adjacent side is the distance along the x-axis

30. anonymous

do i use the pythagorean theorem

31. phi

we could, but we could use cos 30 = x people generally memorize this value

32. anonymous

what would that equal

33. 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.

34. phi

also, drill into your head the idea that cos = x sin = y (for a UNIT circle)

35. 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 $\left( - \frac{\sqrt{3}}{2} , \frac{1}{2} \right)$

36. 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

37. phi