yet another math question i need help with.?

Question

If sin (@ ) = -.3678, how would you find @ in radians in the fourth quadrant?

I think all u do just just take the arcsine of -.3678, and then add 2pi to that answer, but im not sure why u would add 2pi. Any help would be great. Thanks!!

Answer 1

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Hi,

With your calculator set on radians, type in sin^-1(-.3678) and press ENTER. Your answer will be -.3766 radians. What this tells you is that the reference angle measured from the x axis in any quadrant will be .3766 radians. Since the sine value is negative, the angle could be either in the third or fourth quadrant. For the third quadrant the angle would be the x axis of Pi radians + .3766 for a total of 3.5182 radians.
Likewise for the fourth quadrant the angle is measured from 2 Pi radians minus the .3766 radians to put it back into the 4th quadrant. So 2Pi = 6.2832 - .3766 = 5.9066 radians.

To double check these results, take the sin(3.5182) or sin(5.9066). Both should give you a decimal value close to -.3678.

I wrote this as the answer to your earlier question, but I'll enter it here too! I hope that it helps!!

Answer 2

It depends on the domain the teacher wants. arcsin x returns the unique value of y such that sin y = x and -π ≤ y ≤ π. Now here the fact that we're in the fourth quadrant is not a problem, since the fourth quadrant lies within the range of the arcsin function, so the angle arcsin (-.3678) will be in the right quadrant. However, there are infinitely many angles that lie in the fourth quadrant whose sin is -.3678, all of which are separated by an integer multiple of 2π. These are said to be coterminal angles, so named because if you just look at the ray you get by rotating the positive x-axis by any of these angles, it will end up in the same position. Since coterminal angles are geometrically equivalent (in that they end on the same ray), we normally just choose some convenient interval of length 2π in which to do all our measurements, so we don't have to deal with infinitely many redundant representations. In this problem, it seems your teacher has chosen the interval [0, 2π) (as opposed to the slightly more popular (-π, π] ). Since the arcsin function, by default, returns an angle outside that interval when x is negative, you need to add 2π to it to find the angle coterminal to it in [0, 2π) that your teacher wants. But note that the result arcsin (-.3678) isn't incorrect, it IS in the fourth quadrant, it's just your teacher prefers positive angles instead of negative ones.

Answer 3

Draw the sine curve.
-0.3678 can be in the third or fourth quadrant
In the fourth quadrant it is a negative angle, ie clockwise
To get a positive result subtract from a full circle or add 2 Pi.

(Finding the angle in the third quadrant is a bit trickier.)

Answer 4

Take arcsin of .3678 getting .37664207 radians.
The since sin @ = - sin (-@) we get:
@ = -.37664207 radians, or if you like,
@ = 2pi - .37664207 radians

Answer 5

You're right, add 2pi. The reason you might want to add 2pi is to rename the angle as a positive one. It's kind of like calling -90 degrees, 270 degrees.