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camicat

Member

Registered: 2020-02-27

Posts: 11

Expressing Trig functions in terms of other Trig functions

Hello all,

I am struggling with this question:

7. What is 1/cot(x) in terms of sine?

Here is what I have so far, please tell me if I'm starting off incorrectly!

1/cot(x) = tan
tan = sin/cos
1/cot = sin/cos

And then I'm not sure where to go from here. I am supposed to be using the basic identities listed below to help me express 1/cot in terms of sine:

cscΘ = 1/sinΘ
secΘ = 1/cosΘ
cotΘ = 1/tanΘ

sinΘ = 1/cscΘ
cosΘ = 1/secΘ
tanΘ = 1/cotΘ

sinΘ * cscΘ = 1
cosΘ * secΘ = 1
tanΘ * cotΘ = 1
Ratio Identities

sine = opposite / hypotenuse
cosine = adjacent / hypotenuse
tangent = opposite / adjacent

Here is an example of one problem solved correctly in the way my teacher wants me to do it:

11. Express tangent in terms of cosine.

secΘ = 1/cosΘ

->Tan^2Θ + 1 = sec^2Θ

->Tan^2Θ = sec^2Θ – 1

->Tan^2Θ = (secΘ +1) (secΘ -1)

-> Tan^2Θ = (1/cosΘ +1) (1/cosΘ -1)

-> Tan^2Θ = ([1/cosΘ] +[cos Θ /cos Θ]) ([1/cosΘ] –[cos Θ/cos Θ])

-> Tan^2Θ = (1+cosΘ /cosΘ) (1 -cosΘ /cosΘ)

-> Tan^2Θ = (1-cos^2Θ /cosΘ)

-> √Tan^2Θ = √(1-cos^2Θ /cosΘ)

-> tanΘ = √(1-cos^2Θ) /cosΘ

Tangent in terms of cosine is tanΘ = √(1-cos^2Θ) /cosΘ

I hope that someone could give me some direction on where to go from here. We did this earlier in the year and now I am having to solve this problem as part of a review and I just can't remember how to do it. Thanks everyone

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Expressing Trig functions in terms of other Trig functions

hi camicat

In Q11 you have  made use of the trig. identity:

There are several versions of this.  They come directly from Pythagoras Theorem.  If you scroll down this page:

https://www.mathsisfun.com/algebra/trig … ities.html

you'll find them just after the place where the theorem is first mentioned.

So from 1/cot = sin/cos you can express cos in terms of sin and you're done.

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: Expressing Trig functions in terms of other Trig functions

Hi Bob,

I think I understand! Here is what I did:

1/cot = sin/cos

sin²θ + cos²θ = 1     (a trig identity I learned in class)
-> cos²θ = 1 - sin²θ
-> cosθ = √(1 - sin²θ)

So therefore

sin/cos = √(1 - sin²θ)
So,
1/cot = √(1 - sin²θ)

Did I do this right?

I also have another question that I am confused about.

14. Find the principal value of Arctan (-1.44) to the nearest minute.

I have NO clue where to start on this, I don't even know what a principal value is. Could you maybe show me an example of how to solve a question similar to this so that I see the steps of how to find a principal value?

Thank you!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Expressing Trig functions in terms of other Trig functions

hi camicat

You had a fraction

   .....(1)

and then it's gone!

   ......(2)

is correct so substitute (2) into (1) and you will still have a fraction.

Trig functions are periodic so that, for example sin(370) = sin (360 + 10) = sin(10)

So when working with inverse trig, such as arctan, there are many angles that would all give rise to tan(angle) = -1.44

So mathematicians have introduced the idea of principal values to make the inverse into proper functions ie. a single input has just one output.

The domain of the principal values is given here:

https://www.mathsisfun.com/algebra/trig … s-tan.html

You'll have to scroll almost to the bottom of the page but you'll find graphs that show the inverse trig functions (If a 'mapping' has multiple values it isn't called a function).

When you use a calculator, as it has to give a single value, it will have been set up to give the principal value.  For arctangent the domain is -90 to + 90.

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

Mark Dater

Member

Registered: 2021-07-23

Posts: 12

Re: Expressing Trig functions in terms of other Trig functions

Did you use a calculator to calculate?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: Expressing Trig functions in terms of other Trig functions

Yes.

---

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