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# sec, cosec and cot question

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1. Hi guys
Wanting to cover some C3 in the summer, and have come across this question which I can not seem to do, was wondering if someone could give me a push in the right direction?

Simplify cosec(pi/2 - x)
I've learnt that the next step is writing what cosec x = 1/sin x (How would someone prove this? the book doesn't ...)

thus :
cosec(pi/2 - x) = 1/sin (pi/2 - x)

But what else could I do to simply further?

P.s Could someone please clarify some of the terminology to me please? I believe an inverse function is when you reflect a function in y=x? so the inverse of y=sin x is y=sin^-1 x ? However Sin^-1 x =/= 1/sin x ? so Cosec is not an inverse function of sin x?
2. (Original post by coolstorybrother)
cosec x = 1/sin x (How would someone prove this? the book doesn't ...)
It's the definition - literally, cosec is just a quick way of writing 1/sin

(Original post by coolstorybrother)
thus :
cosec(pi/2 - x) = 1/sin (pi/2 - x)

But what else could I do to simply further?

(Original post by coolstorybrother)
P.s Could someone please clarify some of the terminology to me please? I believe an inverse function is when you reflect a function in y=x? so the inverse of y=sin x is y=sin^-1 x ? However Sin^-1 x =/= 1/sin x ? so Cosec is not an inverse function of sin x?
Yeah that's right. The "-1" terminology is confusing. sin^-1 is the inverse function, but cosec is the multiplicative inverse of sin. Just like division is the inverse of multiplication, but the reciprocal 1/x is the multiplicative inverse of x.

A better example is . The inverse function is , but the multiplicative inverse is
3. (Original post by dantheman1261)
It's the definition - literally, cosec is just a quick way of writing 1/sin

aah okay

hhmm I thought it wasn't something straight forward, I have not covered that yet and it wasn't in my book, could you please explain it?

Yeah that's right. The "-1" terminology is confusing. sin^-1 is the inverse function, but cosec is the multiplicative inverse of sin. Just like division is the inverse of multiplication, but the reciprocal 1/x is the multiplicative inverse of x.

A better example is . The inverse function is , but the multiplicative inverse is

I think i get it:
sin^-1 x = inverse
cosec,sec,cot= 1/sin , 1/cos , 1/tan = reciprocal functions?
4. (Original post by coolstorybrother)

hhmm I thought it wasn't something straight forward, I have not covered that yet and it wasn't in my book, could you please explain it?
Ahh - actually, sin(pi/2 - x) can be immediately rewritten as a different trigonometric function (I can't be any more clear without totally giving it away )

(Original post by coolstorybrother)

I think i get it:
sin^-1 x = inverse
cosec,sec,cot= 1/sin , 1/cos , 1/tan = reciprocal functions?
That's it
5. (Original post by dantheman1261)
Ahh - actually, sin(pi/2 - x) can be immediately rewritten as a different trigonometric function (I can't be any more clear without totally giving it away )

That's it
aaaaaah clever! Is it just a translation of sine which makes sin(pi/2 - x) = cosine? Damn, that's a good question.
6. (Original post by coolstorybrother)
aaaaaah clever! Is it just a translation of sine which makes sin(pi/2 - x) = cosine? Damn, that's a good question.
You do not really need to consider the transformation

sin(90-x) = cos(x)

cos(90-x) = sin(x)

Just from the triangles

(used degrees to avoid needing latex)
7. (Original post by coolstorybrother)
aaaaaah clever! Is it just a translation of sine which makes sin(pi/2 - x) = cosine? Damn, that's a good question.
That's right
8. sec x is the inverse of cos x. BECAUSE, LOOK AT THE LETTER C, WHICH INDICATES COS X
COSEC X IS THE INVERSE OF SIN X. BECAUSE, 1/sinx=cosecx or letter S INDICATES SIN X!
So sin(pi/2-x)=cos(2-x)
So cosec(pi/2 - x)=sec(2 - x)
9. (Original post by ECONMATHSMATHSMATH)
sec x is the inverse of cos x. BECAUSE, LOOK AT THE LETTER C, WHICH INDICATES COS X
COSEC X IS THE INVERSE OF SIN X. BECAUSE, 1/sinx=cosecx or letter S INDICATES SIN X!
So sin(pi/2-x)=cos(2-x)
So cosec(pi/2 - x)=sec(2 - x)
I think You should to use the multiplicative inverse or more the reciprocal
term for above.
For functions the inverse, maybe inverse relation or inverse function, and this is another business.
10. (Original post by coolstorybrother)

I think i get it:
sin^-1 x = inverse
cosec,sec,cot= 1/sin , 1/cos , 1/tan = reciprocal functions?
I know loads of books use as the inverse function but yes it is ambiguous. In the end, why would but

Instead, use as the inverse functions!

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