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camicat

Member

Registered: 2020-02-27

Posts: 11

Rewriting Trigonometric Functions

Hello, it's been a long time since I posted last! I am very stuck on this, and could use some help.

I have been tasked with this question:

12. Express cosecant in terms of tangent.

I think I might have a start, but I'm not sure if I'm correct. Here is what I have so far:

tan = sin/cos
sin= 1/csc
tan= (1/csc)/cos

That's as far as I can think. I also have some other, similar problems that I'm struggling with.

13. Express sine in terms of cotangent.

14. Express cosecant in terms of cosine.

15. Express secant in terms of sine.

Any help would be greatly appreciated

Last edited by camicat (2021-03-30 02:17:05)

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,472

Re: Rewriting Trigonometric Functions

Hi camicat,

Welcome to the forum!

See the links in MathsIfFun

Introduction to Trigonometry

and

Trigonometric identities.

Hope these solve the problems!

---

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Rewriting Trigonometric Functions

hi camicat

In a right angled triangle and using A for adjacent, O for opposite and H for hypotenuse:

Pythagoras: O^2 + A^2 = H^2

Divide by H^2:   sin^2 (x) + cos^2 (x) = 1  **

This useful identity can be used for many things including all of your tasks.

eg.

Divide by sin^2 (x) : 1 + 1/tan^2(x) = cosec^2 (x)

** For this to work x has to be an angle in a right angled triangle. But trig values beyond  the range 0-90, are defined in such a way that the result still holds.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

camicat

Member

Registered: 2020-02-27

Posts: 11

Re: Rewriting Trigonometric Functions

Hi Bob,
I'm still very confused. This is how my teacher wants me to look at it (and how my answer should be formatted):

write cosine in terms of tangent, in a way that avoids using sine. One of the other ways to write cosΘ is to say 1/secΘ

We know that tan2Θ + 1 = sec2Θ

This means that secΘ = √(tan2Θ + 1)

So cosΘ = 1/√(tan2Θ + 1)

Putting this together with what we got before we wind up with an answer:

sinΘ = tanΘ * 1/√(tan2Θ + 1)

Can you help me understand how to figure out my answer in that format?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Rewriting Trigonometric Functions

hi camicat

The formulas

and

are really just re-arrangements of the same result.  Let's go back to the right angled triangle.

If you divide through by H squared you get:

If you divide this formula by cos squared you get

So it's 'in the spirit' of what your teacher has told you to use any of the many versions of the formula to help with these problems.

eg. 12. Express cosecant in terms of tangent.

cosecant is 1/sin so I need a version of the formula that has 1/sin in it.

So I'll start with formula (1) and divide through by sin squared.

A full set of identities is on this page: https://www.mathsisfun.com/algebra/trig … ities.html about 2/3 of the way down the page.

Hope that clears it up for you,

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

camicat

Member

Registered: 2020-02-27

Posts: 11

Re: Rewriting Trigonometric Functions

Hi,
That helped a lot! Thank you! I have one last thing that I am stuck on. Here is the problem:

20. Verify (sinΘ)^4 + 2(sinΘ)^2(cosΘ)^2 + (cosΘ)^4 = tanΘcotΘ

I have factored it down so that (sinΘ)4 + 2(sinΘ)2(cosΘ)2 + (cosΘ)4 = (sinΘ^2+cosΘ^2)^2, I'm just not sure where to go from here.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Rewriting Trigonometric Functions

As you know that

all you need is

But one is the reciprocal of the other so that follows straight away.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: Rewriting Trigonometric Functions

As you know that

all you need is

But one is the reciprocal of the other so that follows straight away.

Bob

Hello Bob. I want to learn how to write in LaTex form. Do you have a link in this regard?

Thanks.

Mathegocart

Member

Registered: 2012-04-29

Posts: 2,226

Re: Rewriting Trigonometric Functions

As you know that

all you need is

But one is the reciprocal of the other so that follows straight away.

Bob

Hello Bob. I want to learn how to write in LaTex form. Do you have a link in this regard?

Thanks.

Hey Mathland,

It's a stickied thread in this section.

---

The integral of hope is reality.
May bobbym have a wonderful time in the pearly gates of heaven.
He will be sorely missed.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Rewriting Trigonometric Functions

There are some Latex commands that are not implemented on this forum but most are.  That thread shows what is possible.  You can also construct an expression using this:

https://latex.codecogs.com/legacy/eqneditor/editor.php

Each line must start with:

[math]

and end with a similar /math command. 

I'm having trouble getting this bit to display properly so I hope you can read this ok.

If someone has used Latex in a post and you want to see their commands, just click on the Latex.  You can copy and paste the commands and then edit as you need for your own post.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob