Conversion of a trigonometic function

  • Thread starter frensel
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In summary, the conversation discusses how to convert the trigonometric function \tan(x)\sin(\frac{x}{2})+\cos(\frac{x}{2}) to \frac{\tan(x)}{\sqrt{2(1-\cos(x))}}. The solution involves using angle sum and difference identities, double-angle formula, and half-angle formula to simplify the expression. However, the process is quite complicated and there may be easier ways to convert the function.
  • #1
frensel
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Homework Statement


How to convert
[itex]\tan(x)\sin(\frac{x}{2})+\cos(\frac{x}{2})[/itex]
to
[itex]\frac{\tan(x)}{\sqrt{2(1-\cos(x))}}[/itex]


Homework Equations





The Attempt at a Solution


I can convert it to this form: [itex]\frac{\cos(\frac{x}{2})}{\cos(x)}[/itex]
[itex]\tan(x)\sin(\frac{x}{2})+\cos(\frac{x}{2})[/itex]
=[itex]\frac{\sin(x)}{\cos(x)}\sin(\frac{x}{2})+ \cos(\frac{x}{2})[/itex]
=[itex]\frac{1}{\cos(x)}\left(\sin(x)\sin(\frac{x}{2})+ \cos(x)\cos(\frac{x}{2})\right)[/itex]
using angle sum and difference identities, we get
[itex]\left(\sin(x)\sin(\frac{x}{2})+ \cos(x)\cos(\frac{x}{2})\right) = \cos(x - \frac{x}{2}) = \cos(\frac{x}{2})[/itex]
therefore, we have
[itex]\tan(x)\sin(\frac{x}{2})+\cos(\frac{x}{2}) = \frac{\cos(\frac{x}{2})}{\cos(x)}[/itex]
 
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  • #2
hi frensel :smile:
frensel said:
I can convert it to this form: [itex]\frac{\cos(\frac{x}{2})}{\cos(x)}[/itex]

hint: multiply by sin(x)/sin(x), and use the half-angle identities :wink:
 
  • #3
tiny-tim said:
hi frensel :smile:

hint: multiply by sin(x)/sin(x), and use the half-angle identities :wink:
I got it, thx!

[itex]\frac{\cos(\frac{x}{2})}{\cos{x}}[/itex]
[itex] = \frac{\sin(x)}{\sin(x)}\frac{\cos(\frac{x}{2})}{ \cos{x}}[/itex]
[itex]=\tan(x)\frac{\cos(\frac{x}{2})}{\sin(x)}[/itex]

using double-angle formula, we have
[itex]\tan(x)\frac{\cos(\frac{x}{2})}{\sin(x)}[/itex]
[itex]=\tan(x)\frac{\cos(\frac{x}{2})}{2\sin(\frac{x}{2})\cos(\frac{x}{2})}[/itex]
[itex]=\tan(x)\frac{1}{2\sin(\frac{x}{2})}[/itex]

finally, using half-angle formula (assuming [itex]\sin(\frac{x}{2})>0[/itex]), then

[itex]\tan(x)\frac{1}{2\sin(\frac{x}{2})}[/itex]
[itex]=\tan(x)\frac{1}{2\sqrt{\frac{1-\cos(x)}{2}}}[/itex]
[itex]=\frac{\tan(x)}{\sqrt{2(1-\cos(x))}}[/itex]

Well, although I get the correct result, the calculation is so complicated. Is there any easier way to convert the above trigonometric function?
 
  • #4
you could work backwards (from the answer) …

(tan / 2sin1/2) - cos1/2 = … ? :smile:
 

What is the conversion process for a trigonometric function?

The conversion process for a trigonometric function involves rewriting the function in terms of another trigonometric function or using trigonometric identities to simplify the function.

Why do we need to convert a trigonometric function?

Converting a trigonometric function can make it easier to solve or graph, or it may be necessary to convert a function in order to use a specific trigonometric formula or identity.

What are the most commonly used trigonometric conversions?

The most commonly used trigonometric conversions are those involving sine, cosine, and tangent functions, such as converting sine to cosine or tangent to secant.

How do I know when to use a trigonometric conversion?

You can use a trigonometric conversion when you encounter a function that cannot be easily solved or graphed in its current form, or when you need to use a specific trigonometric formula or identity to solve a problem.

Are there any rules or guidelines for converting trigonometric functions?

Yes, there are several rules and guidelines for converting trigonometric functions, such as using the Pythagorean identities, using the unit circle, and recognizing common patterns in trigonometric functions.

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