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How are Trignometric Ratios related to a circle?
$begingroup$
How can an angle be compared to a circle?
tan means Tangent what does the ratio of Opposite side to Adjacent side have to do with the tangent of a circle?
Similarly, sec means Secant; How does ratio of hypotenuse to Adjacent side have to do with the secant of a circle?
Trigonometric ratios all apply only to a right angled triangle but still the names of the ratios suggest relations to entities related to circles.
I don't get the point of naming the ratios in such a way.
geometry trigonometry
asked Jan 25 at 14:58
YourPalNuravYourPalNurav
1012
$endgroup$
add a comment |
$begingroup$
How can an angle be compared to a circle?
tan means Tangent what does the ratio of Opposite side to Adjacent side have to do with the tangent of a circle?
Similarly, sec means Secant; How does ratio of hypotenuse to Adjacent side have to do with the secant of a circle?
Trigonometric ratios all apply only to a right angled triangle but still the names of the ratios suggest relations to entities related to circles.
I don't get the point of naming the ratios in such a way.
geometry trigonometry
asked Jan 25 at 14:58
YourPalNuravYourPalNurav
1012
$endgroup$
1
$begingroup$
I recommend watching this video <youtube.com/watch?v=6-tsSJx-P18>.
$endgroup$
– Gnumbertester
Jan 25 at 15:05
add a comment |
$begingroup$
How can an angle be compared to a circle?
tan means Tangent what does the ratio of Opposite side to Adjacent side have to do with the tangent of a circle?
Similarly, sec means Secant; How does ratio of hypotenuse to Adjacent side have to do with the secant of a circle?
Trigonometric ratios all apply only to a right angled triangle but still the names of the ratios suggest relations to entities related to circles.
I don't get the point of naming the ratios in such a way.
geometry trigonometry
asked Jan 25 at 14:58
YourPalNuravYourPalNurav
1012
$endgroup$
How can an angle be compared to a circle?
tan means Tangent what does the ratio of Opposite side to Adjacent side have to do with the tangent of a circle?
Similarly, sec means Secant; How does ratio of hypotenuse to Adjacent side have to do with the secant of a circle?
Trigonometric ratios all apply only to a right angled triangle but still the names of the ratios suggest relations to entities related to circles.
I don't get the point of naming the ratios in such a way.
geometry trigonometry
geometry trigonometry
asked Jan 25 at 14:58
YourPalNuravYourPalNurav
1012
asked Jan 25 at 14:58
YourPalNuravYourPalNurav
1012
asked Jan 25 at 14:58
YourPalNuravYourPalNurav
1012
asked Jan 25 at 14:58
YourPalNuravYourPalNurav
1012
asked Jan 25 at 14:58
YourPalNuravYourPalNurav
1012
1012
1
$begingroup$
I recommend watching this video <youtube.com/watch?v=6-tsSJx-P18>.
$endgroup$
– Gnumbertester
Jan 25 at 15:05
add a comment |
1
$begingroup$
I recommend watching this video <youtube.com/watch?v=6-tsSJx-P18>.
$endgroup$
– Gnumbertester
Jan 25 at 15:05
1
1
$begingroup$
I recommend watching this video <youtube.com/watch?v=6-tsSJx-P18>.
$endgroup$
– Gnumbertester
Jan 25 at 15:05
$begingroup$
I recommend watching this video <youtube.com/watch?v=6-tsSJx-P18>.
$endgroup$
– Gnumbertester
Jan 25 at 15:05
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Trigonometric ratios all apply only to a right angled triangle
This statement is not historically accurate.
Historically, trigonometric ratios came from a circle.
The "triangle" part was a later idea.
See this answer for more details.
Somewhere during the history of teaching trigonometry, someone apparently got the idea that only certain triangles you can draw inside the circle are worth teaching to beginning students, and they decided to ignore the circle itself entirely.
I think this is a shame, because it later becomes a stumbling block if you get to the kind of mathematics where you need to take functions of any angle
(including negative angles and angles that "wrap around" multiple times),
not just angles between zero and a right angle.
A reasonably good definition of trigonometric functions can still be based on a circle,
for example, as in graphical representation of trig functions.
There are yet other definitions in higher mathematics that use neither circles nor triangles,
but the circle definition is the one that seems to be the source of the names of the functions.
answered Jan 25 at 16:16
David KDavid K
55.2k344120
$endgroup$
add a comment |
$begingroup$
The length of a line segment tangential to a circle around a central angle is the tangent of that central angle. Look at this graph of the tangent function. 
As you approach each new period, $frac{pi}{2}$, $frac{3pi}{2}$, etc, the value of tangent increases without bound, just as it would if you were to increase the angle $theta$ towards $frac{pi}{2}$ radians or $90$ degrees on this unit circle. 
answered Jan 25 at 15:19
GnumbertesterGnumbertester
670114
$endgroup$
add a comment |
$begingroup$
The origin of tangent is obvious: the function denotes the length intercepted by the (extended) radius along the tangent at point $(1,0)$.
answered Jan 25 at 16:32
Yves DaoustYves Daoust
131k676229
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add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Trigonometric ratios all apply only to a right angled triangle
This statement is not historically accurate.
Historically, trigonometric ratios came from a circle.
The "triangle" part was a later idea.
See this answer for more details.
Somewhere during the history of teaching trigonometry, someone apparently got the idea that only certain triangles you can draw inside the circle are worth teaching to beginning students, and they decided to ignore the circle itself entirely.
I think this is a shame, because it later becomes a stumbling block if you get to the kind of mathematics where you need to take functions of any angle
(including negative angles and angles that "wrap around" multiple times),
not just angles between zero and a right angle.
A reasonably good definition of trigonometric functions can still be based on a circle,
for example, as in graphical representation of trig functions.
There are yet other definitions in higher mathematics that use neither circles nor triangles,
but the circle definition is the one that seems to be the source of the names of the functions.
answered Jan 25 at 16:16
David KDavid K
55.2k344120
$endgroup$
add a comment |
$begingroup$
Trigonometric ratios all apply only to a right angled triangle
This statement is not historically accurate.
Historically, trigonometric ratios came from a circle.
The "triangle" part was a later idea.
See this answer for more details.
Somewhere during the history of teaching trigonometry, someone apparently got the idea that only certain triangles you can draw inside the circle are worth teaching to beginning students, and they decided to ignore the circle itself entirely.
I think this is a shame, because it later becomes a stumbling block if you get to the kind of mathematics where you need to take functions of any angle
(including negative angles and angles that "wrap around" multiple times),
not just angles between zero and a right angle.
A reasonably good definition of trigonometric functions can still be based on a circle,
for example, as in graphical representation of trig functions.
There are yet other definitions in higher mathematics that use neither circles nor triangles,
but the circle definition is the one that seems to be the source of the names of the functions.
answered Jan 25 at 16:16
David KDavid K
55.2k344120
$endgroup$
add a comment |
$begingroup$
Trigonometric ratios all apply only to a right angled triangle
This statement is not historically accurate.
Historically, trigonometric ratios came from a circle.
The "triangle" part was a later idea.
See this answer for more details.
Somewhere during the history of teaching trigonometry, someone apparently got the idea that only certain triangles you can draw inside the circle are worth teaching to beginning students, and they decided to ignore the circle itself entirely.
I think this is a shame, because it later becomes a stumbling block if you get to the kind of mathematics where you need to take functions of any angle
(including negative angles and angles that "wrap around" multiple times),
not just angles between zero and a right angle.
A reasonably good definition of trigonometric functions can still be based on a circle,
for example, as in graphical representation of trig functions.
There are yet other definitions in higher mathematics that use neither circles nor triangles,
but the circle definition is the one that seems to be the source of the names of the functions.
answered Jan 25 at 16:16
David KDavid K
55.2k344120
$endgroup$
Trigonometric ratios all apply only to a right angled triangle
This statement is not historically accurate.
Historically, trigonometric ratios came from a circle.
The "triangle" part was a later idea.
See this answer for more details.
Somewhere during the history of teaching trigonometry, someone apparently got the idea that only certain triangles you can draw inside the circle are worth teaching to beginning students, and they decided to ignore the circle itself entirely.
I think this is a shame, because it later becomes a stumbling block if you get to the kind of mathematics where you need to take functions of any angle
(including negative angles and angles that "wrap around" multiple times),
not just angles between zero and a right angle.
A reasonably good definition of trigonometric functions can still be based on a circle,
for example, as in graphical representation of trig functions.
There are yet other definitions in higher mathematics that use neither circles nor triangles,
but the circle definition is the one that seems to be the source of the names of the functions.
answered Jan 25 at 16:16
David KDavid K
55.2k344120
edited Jan 25 at 16:24
answered Jan 25 at 16:16
David KDavid K
55.2k344120
answered Jan 25 at 16:16
David KDavid K
55.2k344120
answered Jan 25 at 16:16
David KDavid K
55.2k344120
55.2k344120
add a comment |
add a comment |
$begingroup$
The length of a line segment tangential to a circle around a central angle is the tangent of that central angle. Look at this graph of the tangent function.
As you approach each new period, $frac{pi}{2}$, $frac{3pi}{2}$, etc, the value of tangent increases without bound, just as it would if you were to increase the angle $theta$ towards $frac{pi}{2}$ radians or $90$ degrees on this unit circle.
answered Jan 25 at 15:19
GnumbertesterGnumbertester
670114
$endgroup$
add a comment |
$begingroup$
The length of a line segment tangential to a circle around a central angle is the tangent of that central angle. Look at this graph of the tangent function.
As you approach each new period, $frac{pi}{2}$, $frac{3pi}{2}$, etc, the value of tangent increases without bound, just as it would if you were to increase the angle $theta$ towards $frac{pi}{2}$ radians or $90$ degrees on this unit circle.
answered Jan 25 at 15:19
GnumbertesterGnumbertester
670114
$endgroup$
add a comment |
$begingroup$
The length of a line segment tangential to a circle around a central angle is the tangent of that central angle. Look at this graph of the tangent function.
As you approach each new period, $frac{pi}{2}$, $frac{3pi}{2}$, etc, the value of tangent increases without bound, just as it would if you were to increase the angle $theta$ towards $frac{pi}{2}$ radians or $90$ degrees on this unit circle.
answered Jan 25 at 15:19
GnumbertesterGnumbertester
670114
$endgroup$
The length of a line segment tangential to a circle around a central angle is the tangent of that central angle. Look at this graph of the tangent function.
As you approach each new period, $frac{pi}{2}$, $frac{3pi}{2}$, etc, the value of tangent increases without bound, just as it would if you were to increase the angle $theta$ towards $frac{pi}{2}$ radians or $90$ degrees on this unit circle.
answered Jan 25 at 15:19
GnumbertesterGnumbertester
670114
answered Jan 25 at 15:19
GnumbertesterGnumbertester
670114
answered Jan 25 at 15:19
GnumbertesterGnumbertester
670114
answered Jan 25 at 15:19
GnumbertesterGnumbertester
670114
670114
add a comment |
add a comment |
$begingroup$
The origin of tangent is obvious: the function denotes the length intercepted by the (extended) radius along the tangent at point $(1,0)$.
answered Jan 25 at 16:32
Yves DaoustYves Daoust
131k676229
$endgroup$
add a comment |
$begingroup$
The origin of tangent is obvious: the function denotes the length intercepted by the (extended) radius along the tangent at point $(1,0)$.
answered Jan 25 at 16:32
Yves DaoustYves Daoust
131k676229
$endgroup$
add a comment |
$begingroup$
The origin of tangent is obvious: the function denotes the length intercepted by the (extended) radius along the tangent at point $(1,0)$.
answered Jan 25 at 16:32
Yves DaoustYves Daoust
131k676229
$endgroup$
The origin of tangent is obvious: the function denotes the length intercepted by the (extended) radius along the tangent at point $(1,0)$.
answered Jan 25 at 16:32
Yves DaoustYves Daoust
131k676229
answered Jan 25 at 16:32
Yves DaoustYves Daoust
131k676229
answered Jan 25 at 16:32
Yves DaoustYves Daoust
131k676229
answered Jan 25 at 16:32
Yves DaoustYves Daoust
131k676229
131k676229
add a comment |
add a comment |
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$begingroup$
I recommend watching this video <youtube.com/watch?v=6-tsSJx-P18>.
$endgroup$
– Gnumbertester
Jan 25 at 15:05