Maths Question

https://www.quora.com/profile/Bravewarrior/p-143671992
Here is the question and its solution. I actually don't get the question itself or the solution. I have watched some videos on youtube on this topic but am still a bit unclear. So if anyone would please explain the question a bit and the solution I would really appreciate it. Thank you!

Reply 1

mqb2766

What do/don't you understand about arcsin/multiple solutions? Rather than just posting model solutions, it would probably help to say what you do understand.

Reply 2

pigeonwarrior

Original post by mqb2766

What do/don't you understand about arcsin/multiple solutions? Rather than just posting model solutions, it would probably help to say what you do understand.

Hello, I drew the graph of y=arcsinx and restricted it to 0 to 1 as it says in the question. Also if arcsink equals alpha then sin inverse k equals alpha. So that means that k is equal to sin of alpha. Question asks for first two positive values of x for which sinx is equal to k. As I said if k is equal to sin of alpha then k is also equal to sinx therefore sinx and sin alpha are equal. no clue how this helps me at all. But this is how far i got. This question is very confusing to me 😟

Reply 3

tonyiptony

Original post by pigeonwarrior

I think you've fallen into the common trap of thinking arcsin(x) is the inverse of sin(x). This is not completely wrong, but is missing an important detail - what is the domain and range of sin(x) and arcsin(x)?

Remember that you can input whatever real x you want into sin(x), but arcsin(x) only outputs one single value. For instance, inputting 0, pi, 2pi, 3pi, -pi... into sin(x) gives you 0, but arcsin(0) = 0 only (i.e. not pi, 2pi, -pi...).

In a way, this question is asking "how do you get all the solutions to, say, sin(x) = 1/3 when your calculator only tells you one of them (by taking arcsin(1/3))". Of course once you've found all the solutions, you can find the first two positive ones.

(edited 1 year ago)

Reply 4

mqb2766

Just to add to tonys description, in
https://www.desmos.com/calculator/qqf7pvro4b
You have arcsin and sin restricted to the relevant domain -pi/2..pi/2. Sin is increasing on this domain so 1 to 1, so it is invertible (arcsin) on that domain.

In a sense you could pick any similar "pi interval" though it would be very strange if it didn't include 0..pi/2. Then its just a case of adding on the pi/2 interval, before or after, where the trig function is 1 to 1.