sec, cosec and cot question

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?

Reply 1

dantheman1261

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?

Use the sin addition formulae :smile:

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

f(x) = x^3

. The inverse function is

f^{-1}(x) = x^{1/3}

, but the multiplicative inverse is

(f(x))^{-1} = \frac{1}{x^3}

(edited 11 years ago)

Reply 2

coolstorybrother

Original post by dantheman1261

It's the definition - literally, cosec is just a quick way of writing 1/sin

aah okay :smile:

Use the sin addition formulae :smile:

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

f(x) = x^3

. The inverse function is

f^{-1}(x) = x^{1/3}

, but the multiplicative inverse is

(f(x))^{-1} = \frac{1}{x^3}

I think i get it:
sin^-1 x = inverse
cosec,sec,cot= 1/sin , 1/cos , 1/tan = reciprocal functions?

Reply 3

dantheman1261

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 :smile:

)

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

Reply 4

coolstorybrother

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 :smile:

)

That's it :smile:

aaaaaah clever! Is it just a translation of sine which makes sin(pi/2 - x) = cosine? Damn, that's a good question.

Reply 5

TenOfThem

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)

Reply 6

dantheman1261

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 :smile:

Reply 7

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)

Reply 8

ztibor

Original post by ECONMATHSMATHSMATH

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.