Concerning equivalence of sini (sinus pl.(?))

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Discussion Overview

The discussion revolves around the properties of the sine function, particularly in relation to angle equivalence and the implications of sine values being equal for different angles. Participants explore whether equal sine values imply equal angles and the relationships between sine and cosine functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions if sin(x) = sin(x') implies sin(x - 90°) = sin(x' - 90°) = cos(x) = cos(x').
  • Another participant asserts that equal sine values do not necessarily mean the angles are the same, referencing periodicity and identities of the sine function.
  • Some participants express confusion about the relevance of trigonometric identities to their questions about sine values and angle relationships.
  • There is a reiteration that sine is a periodic function, allowing different angles to have the same sine value.
  • One participant notes specific angle relationships, such as sin(x) = sin(π - x) and sin(x) = sin(2π - x), suggesting these identities are relevant to the discussion.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether equal sine values imply equal angles. There are multiple competing views regarding the implications of sine equivalence and the role of trigonometric identities.

Contextual Notes

Participants reference the periodic nature of sine and cosine functions, but there is uncertainty regarding the specific conditions under which sine values can be equal for different angles. The discussion includes references to trigonometric identities without fully resolving their implications for the original question.

Master replies:
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If sin(x)=sin(x') ,where x is some angle and x' a angle of another triangle, does it then follow that sin(x-90°)=sin(x'-90°)=cos(x)=cos(x')?
 
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More generally if the sinus of a angle is equivalent to another does that mean the angles are the same?
 
Master replies: said:
More generally if the sinus of a angle is equivalent to another does that mean the angles are the same?
Not necessarily. There are identities covering for which angles the sine (and all the other trig functions) has the same value. If you don't want to look these up, you can draw the graph of y = sin (x) and pick them out (remember, sine is a periodic function).

BTW, the plural of "sine" is "sines" in English.
 
So is this true: sin(x)=sin(x')→sin(90°-x)=sin(90°-x')=cos(x)=cos(x')?
 
Master replies: said:
So is this true: sin(x)=sin(x')→sin(90°-x)=sin(90°-x')=cos(x)=cos(x')?
Like I said, you can look up the trig identities spelling out the details.

Google "trig identities"
 
I don't quite understand what you mean. If all I know is that sin(x)=sin(x') is it then also true that sin(90-x)=sin(90°-x'). I am sure it is. Do sines only differ by angle, as they only depend on the angle? Sorry if I repeat myself.
 
Master replies: said:
I don't quite understand what you mean. If all I know is that sin(x)=sin(x') is it then also true that sin(90-x)=sin(90°-x'). I am sure it is. Do sines only differ by angle, as they only depend on the angle? Sorry if I repeat myself.

It's very simple. Trig identities can be found on the web by looking them up in a search engine, as I have already stated. You are using a computer to communicate with PF, so you can use the search engine to find these trig identities as easily as I can ...
 
No I mean i don't understand what the trig identities have to do with my problem?
 
Master replies: said:
No I mean i don't understand what the trig identities have to do with my problem?
All of the trig functions are periodic, which means that two different angles can have the same sine (or cosine, tangent, etc.).

This can be seen by looking at a graph of any of these functions. The trig identities mentioned several times by SteamKing provide formulas such as ##\sin(x + 2\pi) = \sin(x)## and many others. As advised, make a web search for trig identities.
 
  • #10
But only the angles in that only differ by a multiple of pi they are not compltly different.
 
  • #11
Master replies: said:
But only the angles in that only differ by a multiple of pi they are not compltly different.
What you wrote is unintelligible. The period for some of the trig functions is ##2\pi##, so, for example, ##\sin(x + \pi) \ne \sin(x)##.

they are not compltly different.
What are not completely different?
 
  • #12
For 0 < x < \frac{\pi}{2},\ sin(x)=sin(\pi -x),\ sin(\pi +x)=sin(2\pi -x)
 

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