How do I use an identity to simplify an expression when the coefficients are different?

[Al-Layth]

I have an identity
[math]c=3a-4a^3[/math]
and I need to rewrite this expression:
[math]a^3 + a[/math]in terms of c ONLY

How do I do this? PLS HELP

 

[Dr.Peterson]

I have an identity
[math]c=3a-4a^3[/math]
and I need to rewrite this expression:
[math]a^3 + a[/math]in terms of c ONLY

How do I do this? PLS HELP

Please show us the entire context. If this is a problem you are working on, or one step of your work, show us the problem itself.

Standing alone, it doesn't seem possible, as [imath]c=3a-4a^3[/imath] is not one-to-one, so there will be different values of a corresponding to one value of c, and they will yield different values of [imath]a^3 + a[/imath].

My guess is that something you have omitted changes the nature of the problem (or else you have made a mistake in getting to this point).

 

[Al-Layth]

Please show us the entire context. If this is a problem you are working on, or one step of your work, show us the problem itself.

Standing alone, it doesn't seem possible, as [imath]c=3a-4a^3[/imath] is not one-to-one, so there will be different values of a corresponding to one value of c, and they will yield different values of [imath]a^3 + a[/imath].

My guess is that something you have omitted changes the nature of the problem (or else you have made a mistake in getting to this point).

thanks
I see so its not possible.

I was actually trying to simplify this sum:
[math]\sin^3(x)+\sin(x)[/math]
into a single trig term. So I thought this identity would help:
[math]\sin(3x)=3\sin(x)-4\sin^3(x)[/math]
But I suppose not.

 

[Dr.Peterson]

thanks
I see so its not possible.

I was actually trying to simplify this sum:
[math]\sin^3(x)+\sin(x)[/math]
into a single trig term. So I thought this identity would help:
[math]\sin(3x)=3\sin(x)-4\sin^3(x)[/math]
But I suppose not.

I had guessed the real problem involved triple angles, because of the form of the "identity".

Note that if you know sin(3x), there are actually three different possible values for x, which again will result in three values for your expression.

Now the question is, why do you want to simplify that expression, and what does "simplify" mean in connection with that goal?

Showing the entire problem you're working on is something we ask for in our guidelines, and with good reason.

 

[Al-Layth]

Im not trying to solve a specific problem. Im just trying to work out a way to write sin(x)+sin^3(x) as a single trig term

just in case I need to be able to do that to solve a trig problem in the future. ?

 

[JeffM]

Why do you think that there is a way to turn that into an expression using only one term?

 

[Steven G]

Yes, it can be done if we use the correct definition of a term.
sin(x)+sin^3(x) = sin(x)(1 + sin^2(x)) or = sin(x)(1 +1-cos^2x)=sin(x)(2-cos^2(x))
You can get the original expression which has two terms into an expression that has one term.

 

[JeffM]

Yes, it can be done if we use the correct definition of a term.
sin(x)+sin^3(x) = sin(x)(1 + sin^2(x)) or = sin(x)(1 +1-cos^2x)=sin(x)(2-cos^2(x))
You can get the original expression which has two terms into an expression that has one term.

[imath](2 - \cos^2 x)[/imath] has two terms.

I suspect that, based on the original question, that what the OP is looking for is

[math]\sin(x) + \sin^3(x) = f(y), \text { where } f \text { is a trigonometric function.}[/math]

 

[Dr.Peterson]

Im not trying to solve a specific problem. Im just trying to work out a way to write sin(x)+sin^3(x) as a single trig term

just in case I need to be able to do that to solve a trig problem in the future. ?

I'm guessing that you have been given problems in the past that ask you to simplify an expression into a form that consists of a single trig function (which is done as a way to give you a specific meaning for "simplify", in practice problems, so there is one correct answer), and that gave you the impression that this can be done for any expression, or at least is common.

It can't. That is a very special situation. A sum of two terms looks quite simple to me! Trig gets a whole lot worse.

Even sin(x)-sin^3(x), which you can do a lot more with, can't be written as just a single function (as far as I can see without working on it).

 

[Steven G]

[imath](2 - \cos^2 x)[/imath] has two terms.

That is true and not true.
Terms are separated my addition and subtraction symbols NOT inside parenthesis.

sin(x)(1 + sin^2(x)) is one single term. You need to look at the whole expression before you can determine how many terms there are.

I have a simply question for you which is in every algebra book I have seen.
Identify the terms in 1 + sin(x)(1 + sin^2(x)) - 2xy
How many terms are there?

I also updated my previous post saying that what I ended up with is a single term but not a single trig term.

 

[JeffM]

That is true and not true.
Terms are separated my addition and subtraction symbols NOT inside parenthesis.

sin(x)(1 + sin^2(x)) is one single term. You need to look at the whole expression before you can determine how many terms there are.

I have a simply question for you which is in every algebra book I have seen.
Identify the terms in 1 + sin(x)(1 + sin^2(x)) - 2xy
How many terms are there?

I also updated my previous post saying that what I ended up with is a single term but not a single trig term.

I’ll not debate definitions. My major point was that I did not think the OP meant “term” in a strict sense.

When I tutor I define an expression in algebra as representing a number. A complex expression is an expression that consists of more than one expression joined by operations. So I tend to think of expressions as a sort of tree with different descriptions pertaining to trunk through twigs.