What is sin(1) , not sin(10) and not sin(1c) ?

[phymatter]

what is sin(1) , not sin(1⁰) and not sin(1^c) ?

 

[alxm]

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.

sin(1) would be that ratio for a right-sided triangle where the angle is "1". Since you didn't define what an angle of "1" is supposed to mean (other than rejecting existing measures of angle), I won't either.

 

[phymatter]

Since you didn't define what an angle of "1" is supposed to mean (other than rejecting existing measures of angle), I won't either.

Actually , here by sin (1) , i mean 1 is a natural number , actually I want to know about the validity of this term : sin (1) , where 1 is a natural number .

 

[arildno]

To add to axlm's post:

It is meaningless to ask what sin(1) is, unless you specify the angular measure.

If you say, for example, that "1" is the angular measure where the full circle is given the value "4", then sin(1)=1.


Note therefore, that the sine "function" itself, regarded as a function from the real numbers onto itself, is actually a FAMILY OF FUNCTIONS, where each member of that family is a proper function with respect to a particular choice of angular measure.

When "x" is measured as "degrees", sin(x) is a DIFFERENT function than when "x" is measured in radians.

 

[HallsofIvy]

I disagree with Arildno's statement (I've become braver about that sort of thing!) that "It is meaningless to ask what sin(1) is, unless you specify the angular measure."

Sine and cosine, as defined in higher mathematics (there is another thread here on exactly that question) have nothing to do with angles or triangles- they are simply functions and, like all mathematical functions, their arguments are simply numbers.

Look at what I gave, in that other thread, as a definition of sine and cosine:

"Draw a unit circle on a coordinate system. For non-negative t, start at (1, 0) on the circumference of the circle and measure counter clockwise around the circumference of the circle a distance t. The point at which we end has coordinates, by definition, (cos(t), sin(t))." (If t is negative, measure clockwise.) When I say "by definition" I mean that is the definition of sine and cosine: what ever that point is, the first coordinates is cos(t), the second sin(t).

Notice that t, here, is not a an angle at all! It is a distance measured around the unit circle! Of course, calculators are designed by engineers and engineers tend to think of sine and cosine in terms of angles (You will see the phrase "phase angle" in statements about electrical circuits that have no angles at all!) so they "create" an angle measure to fit: radian measure. Any time you see sine or cosine without any angle units indicated, or, for that matter, any time you see trig functions in problems where there are no angles or triangles, the argument is to be interpreted in "radians".

sin(1)= sin(1 radian)= 0.8414709848078965066525023216303, approximately.

Again, the argument, t, in sin(t), is not an angle at all, it is a number with no units. But to keep our engineer friend happy, we say "radians".

 

[arildno]

Well, I can live with that disagreement, HallsofIvy!

Personally, I would regard the mapping (sin(t),cos(t)) onto the circle as not fully defined until you specify precisely how an interval on the "t"-line maps onto the circle.
(For example, specify how far on the "t"-line you've come when you have made a full circle)


Without that, the definition is fairly under-nourished on "meaning", if not wholly starving..

 

[phymatter]

Notice that t, here, is not a an angle at all! It is a distance measured around the unit circle! Of course, calculators are designed by engineers and engineers tend to think of sine and cosine in terms of angles (You will see the phrase "phase angle" in statements about electrical circuits that have no angles at all!) so they "create" an angle measure to fit: radian measure. Any time you see sine or cosine without any angle units indicated, or, for that matter, any time you see trig functions in problems where there are no angles or triangles, the argument is to be interpreted in "radians".

sin(1)= sin(1 radian)= 0.8414709848078965066525023216303, approximately.

Again, the argument, t, in sin(t), is not an angle at all, it is a number with no units. But to keep our engineer friend happy, we say "radians".

This is a bit confusing , I clearly understand that we can define sine function by a unit circle but by this do you mean that "Radian is a natural number" !

 

[CRGreathouse]

This is a bit confusing , I clearly understand that we can define sine function by a unit circle but by this do you mean that "Radian is a natural number" !

Radian is the natural number 1, just like percent is the rational number 0.01 and degree is the real number pi/180.

 

[g_edgar]

Radian is the natural number 1, just like percent is the rational number 0.01 and degree is the real number pi/180.

A measurement in radians is unitless. We may obtain it as a ratio of two lengths (arc around a circle divided by radius of circle) for example.

 

[HallsofIvy]

Radian is the natural number 1, just like percent is the rational number 0.01 and degree is the real number pi/180.

Well, that's not at all what I said! And I did not use the phrase "natural number"- certainly not in what phymatter quoted. I said, just as g edgar did, that it is a number without units.

f(x)= sin(x) requires no units just as g(x)= x² does not. Strictly speaking the "1" in sin(1) has no units at all- but to keep our engineer friends happy, we can associate that with the angle "1 radian".

Arildno, there are other ways, as I point out in that other thread I mentioned, of defining "sin(x)" and "cos(x)". One way, for example, is to define them in terms of power series, the other as solutions to initial value, differential equation, problems. In neither of those is x interpreted as being a angle- but in order to use the "sin" and "cos" keys on your calculator you have to use the "radian" mode.

I prefer the last because that makes it easier to prove the periodicity (though it is still a chore!) of sine and cosine.

 

[arildno]

I prefer the last because that makes it easier to prove the periodicity (though it is still a chore!) of sine and cosine.

I also prefer that method as the most rigorous.

It's not that easy to understand for a teenager just learning about generalizations of the trig functions, though.