Question

1. csc^-1(csc(-pi/4))=-pi/4
TRUE OR FALSE and explain why
2. sec(sec^-1(sqrt3))=sqrt3
TRUE OR FALSE and explain why

Answer 1

@jtug6

Answer 2

yes and yes

Answer 3

^

Answer 4

Haha, but why?

Answer 5

I have to have an explanation for part of my answer

Answer 6

that's how inverse functions work, really.
csc(-pi/4) gives you the cosecant of -pi/4 (we don't really care what it is)
then we're entering this value into the csc^(-1) function, which tells us the angle that has that cosecant. but we know the angle that has this cosecant: it's -pi/4.

(there are other tidbits, like the inverse cosecant returns the angle in the range -pi/2 to pi/2, but fortunately here they happened to make the problem simple for us.)

Answer 7

Awesome thanks! @ParthKohli

Answer 8

haha no problem, but can you come up with a similar explanation for the second?

Answer 9

Yes, it's is the same except for the inverse function is on the inside of the brackets... so the inverse is taken before the actual function is taken?

Answer 10

yeah, but i hope that explanation above made sense.

Answer 11

Could I essentially say that they cancel each other out?

Answer 12

@ParthKohli

Answer 13

yes, you can think of it that way.
actually that's literally the definition of an inverse function.

but be careful - \(\sin^{-1} (\sin x)\) is not always equal to \(x\) - likewise for other trigonometric functions too.

Answer 14

Okay, awesome! Haha. But why is that not always to x though? @ParthKohli

Answer 15

alright, the reason is this:
there are infinitely many angles that have the same sine, right?
so if I asked you the inverse sine of 1, would you say pi/2? or 5pi/2? or -3pi/2? because all of these angles have their sin equal to 1.

that is why inverse sine has been defined as the angle BETWEEN \(-\pi/2\) and \(\pi/2\) which has the sine equal to the input... this ensures that there is only one angle returned for a given sine!

so could you tell me the value of \(\sin^{-1}(\sin (2\pi/3))\)?

Answer 16

Oh I see! So, it would have to be between that range, therefore, it would be pi/3 because that is the same value?

Answer 17

@ParthKohli

Answer 18

Yes, amazing! So now you see that you don't get the same angle.

Answer 19

Adding to this, can you guess the condition for \(\sin^{-1}(\sin(x))= x \)?

Answer 20

I'm sorry, but what do you mean by the condition? @ParthKohli

Answer 21

Well, for what values of \(x\) does \(\sin^{-1}(\sin (x)) = x\) hold true?
In other words, I'm just asking you to solve the equation.

Answer 22

Oh, I see. It would merely be the range of sin, right?

Answer 23

I also have another question: so for this study guide I am doing I am asked to solve problems such as 2cosx+5=4. That simplifies to cosx=-1/2. So, when I am evaluating x, would I say that the answer would be x=2/3pi +/- 2npi and x=4/3pi +/- 2npi?

Answer 24

Range of sin?

Answer 25

-pi/2 to pi/2? or -1 to 1

Answer 26

-1to 1

Answer 27

You must have meant the range of arcsine.

Answer 28

oh yes!! haha sorry.

Answer 29

I do have a question though, for question 1. Wouldn't it be false because the range of csc is (-infinity, -1) U (1 to infinity) and the value is pi/4?

Answer 30

But we're talking about inverse cosec here.

Answer 31

I'm sorry... so is there a different range for the inverse of arccsc?

Answer 32

I'm so sorry to be taking so much of your time @ParthKohli I really appreciate it!

Answer 33

Yes, it's -pi/2 to pi/2 (excluding zero, of course).

Answer 34

Inverse trig functions return an angle.

Answer 35

Oh great! Ha. I did not know that was a thing. So sorry, but one more thing: I also have another question: so for this study guide I am doing I am asked to solve problems such as 2cosx+5=4. That simplifies to cosx=-1/2. So, when I am evaluating x, would I say that the answer would be x=2/3pi +/- 2npi and x=4/3pi +/- 2npi?

Answer 36

So what I had put down would be the answer?

Answer 37

OK, I'm dumb right now.

Answer 38

Wow. Thank you so very much. I can't tell you how much this has clarified my problems! I pride myself in math, but sometimes you just have those moments where you cannot figure things out (or second guess yourself haha). So seriously, thanks!

Answer 39

The condensed form is \(2n \pi \pm \frac{2\pi}3\). I just confused myself thrice.

Answer 40

hahaha. But my answer will be marked as correct, correct?

Answer 41

yeahhh I guess