SOLUTION: THE TERMINAL ARMS OF ANGLES IN STANDARD POSITION PASS THROUGH THE FOLLOWING POINTS. FIND THE MEASURE OF EACH ANGLE IN RADIANS, TO THE NEAREST HUNDRETH a) (-3,14) b) (6,7)

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Question 372794: THE TERMINAL ARMS OF ANGLES IN STANDARD POSITION PASS THROUGH THE FOLLOWING POINTS. FIND THE MEASURE OF EACH ANGLE IN RADIANS, TO THE NEAREST HUNDRETH
a) (-3,14)
b) (6,7)
Found 2 solutions by jsmallt9, robertb:
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
It may help to draw a picture. An angle in "standard position" has its vertex on the origin and one side of the angle is the positive part of the x axis. The other side, then, starts from the origin and goes through the given point. If you look at your drawing and if you understand your trig ratios then you will see that the ratio of the y coordinate over the x coordinate of the given point will be the tangent of the angle in question.

So for
a) (-3, 14)
the tangent of the angle will be 14/-3. When you have a ratio and you want to figure out the angle, you use inverse functions. On you calculator, the inverse tangent function button will probably look like one of the following:
tan-1
atan
arctan
To find the angle,
  1. Since we want radian measure make sure your calculator is set to radian measure. (If you don't know how to do this, then see "Unknown mode" below.)
  2. enter tan-1(14/3). (Note: We are leaving off the minus from negative 3 for a reason!)
We should get something close to 1.3597029935721501. This is the reference angle. Since (-3, 14) is in the 2nd quadrant the angle itself will be minus the reference angle. (This is where we handle the minus of negative 3.)
So the angle is - 1.3597029935721501

For b) (6, 7)
tan-1(7/6) = 0.8621700546672263
So the reference angle is 0.8621700546672263. Since (6, 7) is in the first quadrant, the reference angle and the angle itself are one and the same. So the answer is 0.8621700546672263

Unknown mode
If you don't know how to change the mode (or if you don't even know what mode your calculator is using) then here is how you can do problems like this:
  1. Determine the current mode of the calculator. To do this enter:
    cos-1)(-1)
    If you get 180, the calculator is in degree mode
    If you get a decimal that looks like , 3.1415926535897931..., then the calculator is in radian mode.
    If you get something else, then you are in some other mode (probably gradient) and you'll have to look up how to convert from that mode into the desired mode. (You're on your own if you are stuck in gradient mode. I forget the conversions for it.)
  2. Find the desired inverse function value.
  3. If the calculator is
    • in degree mode and you want a radian answer, then multiply the result from step 2 by . This will convert an angle in degrees to the angle in radian measure.
    • If the calculator is in radian mode and you want a degree answer, then multiply the result from step 2 by . This will convert an angle in radian measure and convert it to degrees.
Answer by robertb(5830)   (Show Source): You can put this solution on YOUR website!
a) The angle is given as being in standard position already. Let A be the angle. Then . If you're using a scientific calculator, the inverse tan key recognizes only angles coming from the interval (, ), the principal domain of tan. Thus . This angle is in the 4th quadrant. To get it back to the 2nd quadrant, add to this, which is the period of tangent. Thus A = 1.78 radians.
b) Let A be the angle . Then . radians, using a scientific calculator. This is already in quadrant 1, so no more adjustment is needed.
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