# Why is sin the y-coordinate on the unit circle?

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#1
I'm sure my question will be instantly misunderstood by most people in its current form, so I'll elaborate.

Basically, I mean why the specific names of the functions were chosen to be that way around. You would think, intuitively, that we would naturally attribute sine to the x-coordinate and COsine to the y-coordinate, just as we watch how y varies as we vary x on the Cartesian plane (by convention).

It makes sense, though, to have sine be the y-coordinate because of how the graphs look (y=sinx is the 'proper' wave and y=cosx looks like your usual wave shifted along). But is this how the names were chosen? Were the graphs plotted before the names were attributed?

I know this isn't really a mathematical question, but I've looked up the question and couldn't find any answers; I'm very curious to know.
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5 years ago
#2
(Original post by PrimeLime)
I'm sure my question will be instantly misunderstood by most people in its current form, so I'll elaborate.Basically, I mean why the specific names of the functions were chosen to be that way around. You would think, intuitively, that we would naturally attribute sine to the x-coordinate and COsine to the y-coordinate, just as we watch how y varies as we vary x on the Cartesian plane (by convention).It makes sense, though, to have sine be the y-coordinate because of how the graphs look (y=sinx is the 'proper' wave and y=cosx looks like your usual wave shifted along). But is this how the names were chosen? Were the graphs plotted before the names were attributed?I know this isn't really a mathematical question, but I've looked up the question and couldn't find any answers; I'm very curious to know.
Check out this picture ------> (let's call the coordinates of that blue point (x,y))

As you can see, sin(θ) = y/1 = y

This is why the y-coordinate of that blue point of is the sin(θ) value, and this only works with the unit circle.
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5 years ago
#3
(Original post by Dingooose)
Check out this picture ------>

As you can see, sin(θ) = y/1 = y

This is why the y-coordinate is the sin value.
Do I smell Autograph?
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5 years ago
#4
(Original post by Alexion)
Do I smell Autograph?
No, I'm using this: https://www.desmos.com/calculator
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5 years ago
#5
Updated pic:
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5 years ago
#6
(Original post by Dingooose)
No, I'm using this: https://www.desmos.com/calculator
Well that looks significantly easier & nicer to use, and it's free...

I laugh in the face of my maths teachers
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5 years ago
#7
(Original post by Alexion)
Well that looks significantly easier & nicer to use, and it's free...

I laugh in the face of my maths teachers
Yeah, they'd be surprised at how much cool stuff you can get/use for free.
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#8
(Original post by Dingooose)
Check out this picture ------>

As you can see, sin(θ) = y/1 = y

This is why the y-coordinate is the sin value.
I thought someone would answer with this reasoning.

It seems reasonable to say this, but there is a logical flaw in the argument. You are using the definition: sin(theta)=opposite leg / hypotenuse in a right-angled triangle.

But then you are defining the function sin insufficiently. You are only defining it for angles 0 to 90 (degrees), using your right-angled triangle definition. This is precisely why sin is defined via the unit circle definition - so that it is defined for all angles. And then that asks the question, why was sin defined as the y-coordinate of the terminal point?
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5 years ago
#9
(Original post by PrimeLime)
I thought someone would answer with this reasoning.

It seems reasonable to say this, but there is a logical flaw in the argument. You are using the definition: sin(theta)=opposite leg / hypotenuse in a right-angled triangle.

But then you are defining the function sin insufficiently. You are only defining it for angles 0 to 90 (degrees), using your right-angled triangle definition. This is precisely why sin is defined via the unit circle definition - so that it is defined for all angles. And then that asks the question, why was sin defined as the y-coordinate of the terminal point?
TBH I'm seriously confused what you're trying to ask here. Except for the fact that most mathematical functions are defined as y=f(x), hence y=sinx, idekwya
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5 years ago
#10
(Original post by PrimeLime)
I thought someone would answer with this reasoning.

It seems reasonable to say this, but there is a logical flaw in the argument. You are using the definition: sin(theta)=opposite leg / hypotenuse in a right-angled triangle.

But then you are defining the function sin insufficiently. You are only defining it for angles 0 to 90 (degrees), using your right-angled triangle definition. This is precisely why sin is defined via the unit circle definition - so that it is defined for all angles. And then that asks the question, why was sin defined as the y-coordinate of the terminal point?
Khan Academy uses the exact same reasoning as I do. Check out this vid: https://www.khanacademy.org/math/tri...ig-functions-1

The unit circle definitions are an extension of the normal definitions, but they are based upon the normal definitions.
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5 years ago
#11
(Original post by Alexion)
TBH I'm seriously confused what you're trying to ask here. Except for the fact that most mathematical functions are defined as y=f(x), hence y=sinx, idekwya
He's asking why, in the unit circle, if you draw a radius from (0,0) to the circumference of the circle, the sine of the angle created between the radius and the x-axis is the same as the y coordinate of the terminal point (point where radius touches the circumference).
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5 years ago
#12
(Original post by PrimeLime)

Basically, I mean why the specific names of the functions were chosen to be that way around. You would think, intuitively, that we would naturally attribute sine to the x-coordinate and COsine to the y-coordinate, just as we watch how y varies as we vary x on the Cartesian plane (by convention).

It makes sense, though, to have sine be the y-coordinate because of how the graphs look (y=sinx is the 'proper' wave and y=cosx looks like your usual wave shifted along). But is this how the names were chosen? Were the graphs plotted before the names were attributed?

.
Nothing to do with 'graphs' - trig functions were known about long before anyone even thought about graphs (certainly not before Descartes started thinking about coordinates!).

The word 'sine' is just an anglicization of a Latin translation of an Arabic corruption of a Hindu phrase meaning 'half-chord' - a long time ago, Indian scholars compiled tables of half-chord lengths corresponding to different angles at the centre of a circle. This half-chord length is what we recognize today as the y-coordinate of a point on the circle,
3
#13
(Original post by davros)
Nothing to do with 'graphs' - trig functions were known about long before anyone even thought about graphs (certainly not before Descartes started thinking about coordinates!).

The word 'sine' is just an anglicization of a Latin translation of an Arabic corruption of a Hindu phrase meaning 'half-chord' - a long time ago, Indian scholars compiled tables of half-chord lengths corresponding to different angles at the centre of a circle. This half-chord length is what we recognize today as the y-coordinate of a point on the circle,
Ah, that clears things up a bit. Thanks.
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