Finding an expression for (e.g. sin (3x)) in terms of (e.g. sin x) alone?

In summary, the question is asking to find a way to express sin(3x) using only sin(x) without any cosines. This can be done by using the addition formula for sin(2x) and the identity for cos(2x) that involves only sin(x).
  • #1
liquidwater
11
0

Homework Statement


Use double-angle and addition formulæ and other relations for trigonometrical functions to find an expression for sin(3x) in terms of sin x alone.

My problem is I don't know what is meant by "find an expression for sin(3x) in terms of sin x alone.". I know the relevant formulae but do not know what is actually wanted of the question. I know I can 'split' it by going sin(2x + x) then using addition formulae... But I don't know why or what is expected as a final answer. An equation involving only sin and no cos?

Homework Equations


Trig identities, addition formulae


The Attempt at a Solution


No idea.


PS. I found the answer online but had no idea why that's the answer - please don't just give me the answer :).
 
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  • #2
Basically, what the question is asking you is to find another way to write sin(3x) using only sin(x). So your final answer cannot have any sin(3x) or sin(2x), but only sin(x).

I think your first idea is a good one:

[tex]sin(3x) = sin(2x + x) = sin(2x)cos(x) + cos(2x)sin(x)[/tex]

However, this still has sin(2x), cos(2x) and cos(x) in it. How can you get rid of all these and be left with just a combination of sin(x)?
 
  • #3
liquidwater said:

Homework Statement


Use double-angle and addition formulæ and other relations for trigonometrical functions to find an expression for sin(3x) in terms of sin x alone.

My problem is I don't know what is meant by "find an expression for sin(3x) in terms of sin x alone.". I know the relevant formulae but do not know what is actually wanted of the question. I know I can 'split' it by going sin(2x + x) then using addition formulae... But I don't know why or what is expected as a final answer. An equation involving only sin and no cos?

1. Use your idea about addition formulae. Just apply it once.

2. Then consider any formulas for [itex]\sin(2x)[/itex] and [itex]\cos(2x)[/itex]? In particular, you will want the identity for [itex]\cos(2x)[/itex] that involves only [itex]\sin x[/itex] as there are three identities for [itex]\cos(2x)[/itex]. And don't forget the most basic one: [itex](\sin x)^2 + (\cos x)^2 = 1[/itex].
 
  • #4
Doh! Looks like I type too slowly at this early hour... Danago beat me to the punch!
 
  • #5
Thanks a lot to both of you, I actually understand what is required now.

I'm a bit lost with actually getting the solution, but I really do need to work on my math skills so I'll do that.

Thanks again!
 
  • #6
Give it a good shot, and if you get lost in the algebra and trig. identities, feel free to post back here and I am sure someone will be able to help out :smile:

All the best,
Dan.
 
  • #7
Give it a good shot, and if you get lost in the algebra and trig. identities, feel free to post back here and I am sure someone will be able to help out :smile:

All the best,
Dan.
 

Related to Finding an expression for (e.g. sin (3x)) in terms of (e.g. sin x) alone?

1. What does it mean to find an expression for (e.g. sin (3x)) in terms of (e.g. sin x) alone?

Finding an expression for a trigonometric function in terms of another trigonometric function means representing the first function using only the second function. In other words, we want to write the given function (e.g. sin (3x)) as a combination of the second function (e.g. sin x) and possibly some constants or coefficients.

2. Why would we want to find an expression for a function in terms of another function?

There are a few reasons why we might want to do this. One is to simplify a complicated expression involving multiple trigonometric functions into a more concise form. Another reason is to make it easier to graph or evaluate the function, especially if it involves special angles or values.

3. How do we find an expression for (e.g. sin (3x)) in terms of (e.g. sin x) alone?

To find an expression for one trigonometric function in terms of another, we use trigonometric identities, which are equations that relate different trigonometric functions. For example, to find an expression for sin (3x) in terms of sin x, we can use the double angle identity for sine, which states that sin (2x) = 2sin x cos x. We can then rewrite sin (3x) as sin (2x + x) and use the sum identity for sine to get sin (3x) = sin (2x) cos x + cos (2x) sin x. Finally, we can substitute the known expression for sin (2x) to get sin (3x) = (2sin x cos x) cos x + cos (2x) sin x, and use the double angle identity for cosine to simplify further.

4. Are there any general strategies or tips for finding expressions in terms of other functions?

One helpful strategy is to look for patterns and use known identities or formulas to manipulate the given function into the desired form. It can also be helpful to start with simpler expressions and build up to the desired one, using identities and properties of trigonometric functions along the way.

5. Can we always find an expression for a function in terms of another function?

In general, yes, we can find an expression for a function in terms of another function. However, the process may not always be straightforward and may require multiple steps or the use of more complex identities. Additionally, some functions may not have a simple expression in terms of another function, in which case numerical methods may be used to approximate the function.

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