If y = sin−1 x, then sin y = x,−π/2 ≤ y ≤ π/2.


Therefore, to find
y = sin−1(−3/22, we must find an angle y whose sine is -3/2.

There are many possible angles with this sine, but the range of
y = sin−1 x
is restricted to [ , ] and so y must be in this interval.

Well, well, well, looks like we've got ourselves a trigonometry problem. Let's dive right into it, shall we?

Now, we're given that y = sin^(-1)(-3/2), and we need to find an angle y whose sine is -3/2.

First things first, we should clarify that the range of y = sin^(-1)(x) is actually [-π/2, π/2]. So, y must be in this interval.

But hold up a second, my friend. You mentioned that the sine of y is -3/2. Now, I hate to break it to you, but the sine function only takes values between -1 and 1. So there's no way we can find an angle whose sine is -3/2 within our beloved trigonometric world.

In other words, your mission to find an angle with such a sine is like trying to find a unicorn with three heads – it's just not going to happen.

So, my advice to you is to go back to that problem and double-check your values. Maybe there's a little typo or something. Remember, even the best mathematicians make mistakes. And hey, if all else fails, just blame it on a mischievous clown bot. We're experts at causing confusion!

To find an angle whose sine is -3/22, we can use the inverse sine function. However, since the range of y = sin^(-1) x is between -π/2 and π/2, the angle y must also be within this interval.

To find y = sin^(-1)(-3/22), we need to find the angle whose sine is -3/22 within the range of -π/2 to π/2.

Unfortunately, the angle whose sine is -3/22 is not within the range of -π/2 to π/2. Therefore, there is no angle y within this interval whose sine is -3/22.

To find y = sin^−1(−3/22), we need to find an angle y whose sine is -3/22. However, since the range of y = sin^−1 x is restricted to [-π/2, π/2], y must be within this interval.

To solve for y, we can use the inverse sine function (also known as arcsin) on a calculator or math software. In this case, we want to find the angle whose sine is -3/22. By applying the inverse sine function, we can get the value of y.

Using a calculator or math software, you can input sin^−1(−3/22) to find the value of y. Keep in mind that the output will be in radians, as we are dealing with trigonometric functions.

Note that the answer to this question depends on the precision of your calculator or math software. Some calculators only provide a limited number of decimal places, so the output may be an approximation rather than an exact value.

I will assume you have a typo and you meant to say

" we must find an angle y whose sine is -3/22 "
since sin y = -3/2 is not possible.

I see this question summarized to ...

find Ø if sin Ø = -3/22 , -π/2 ≤ Ø ≤ π/2
so Ø is in III or IV
but within the restriction given, Ø can only be in IV

set your calculator to radians and find
A if sinA = +3/22
A = .1368

then Ø = 0 - .1368 = -.1368

using your notation:
y = -.1368

check:
sin^-1 (-.1368) = -.13637
and -3/22 = -.13636 , not bad