How do geometric properties of sine and cosine follow from their power series definition?
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If you define $cos$ and $sin$ using their power series, or as the real and imaginary part of the power series of $e^{ix}$, how can you prove that they are periodic? Also, how do you prove that period is $pi$? And how do you prove that the points $(cos(x), sin(x))$ for $x in [0, 2pi]$ form a circle?
I believe the last question can be proven if you use the continuity of $cos$ and $sin$, which follows from their power series definition, and from the fact that $cos^2(x) + sin^2(x) = 1$, but using only these two properties is not enough for proving they form a full circle I believe. I think you also need to find their derivatives on the intervals $[0, pi/2]$, $[pi/2, pi]$, $[pi, 3pi/2]$ and $[3pi/2, 2pi]$, is this correct? If so, how can this be done?
calculus analysis trigonometry power-series
asked Nov 17 at 20:22
Surzilla
1238
add a comment |
up vote
3
down vote
favorite
If you define $cos$ and $sin$ using their power series, or as the real and imaginary part of the power series of $e^{ix}$, how can you prove that they are periodic? Also, how do you prove that period is $pi$? And how do you prove that the points $(cos(x), sin(x))$ for $x in [0, 2pi]$ form a circle?
I believe the last question can be proven if you use the continuity of $cos$ and $sin$, which follows from their power series definition, and from the fact that $cos^2(x) + sin^2(x) = 1$, but using only these two properties is not enough for proving they form a full circle I believe. I think you also need to find their derivatives on the intervals $[0, pi/2]$, $[pi/2, pi]$, $[pi, 3pi/2]$ and $[3pi/2, 2pi]$, is this correct? If so, how can this be done?
calculus analysis trigonometry power-series
asked Nov 17 at 20:22
Surzilla
1238
The reverse question is asked here: "Deriving the power series for cosine, using basic geometry". You may find my answer interesting.
– Blue
Nov 17 at 20:25
1
@Blue thank you, but I can't quite follow the proof in that answer, especially since the combinatorial argument is omitted. I appreciate the reference and will probably return to it when my math skills are stronger, but for now I'd appreciate a proof in the direction I asked.
– Surzilla
Nov 17 at 21:01
1
Possible duplicate: How to prove periodicity of $sin(x)$ and $cos(x)$ starting from the Taylor seires expansion?
– YiFan
Nov 18 at 10:44
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
If you define $cos$ and $sin$ using their power series, or as the real and imaginary part of the power series of $e^{ix}$, how can you prove that they are periodic? Also, how do you prove that period is $pi$? And how do you prove that the points $(cos(x), sin(x))$ for $x in [0, 2pi]$ form a circle?
I believe the last question can be proven if you use the continuity of $cos$ and $sin$, which follows from their power series definition, and from the fact that $cos^2(x) + sin^2(x) = 1$, but using only these two properties is not enough for proving they form a full circle I believe. I think you also need to find their derivatives on the intervals $[0, pi/2]$, $[pi/2, pi]$, $[pi, 3pi/2]$ and $[3pi/2, 2pi]$, is this correct? If so, how can this be done?
calculus analysis trigonometry power-series
asked Nov 17 at 20:22
Surzilla
1238
If you define $cos$ and $sin$ using their power series, or as the real and imaginary part of the power series of $e^{ix}$, how can you prove that they are periodic? Also, how do you prove that period is $pi$? And how do you prove that the points $(cos(x), sin(x))$ for $x in [0, 2pi]$ form a circle?
I believe the last question can be proven if you use the continuity of $cos$ and $sin$, which follows from their power series definition, and from the fact that $cos^2(x) + sin^2(x) = 1$, but using only these two properties is not enough for proving they form a full circle I believe. I think you also need to find their derivatives on the intervals $[0, pi/2]$, $[pi/2, pi]$, $[pi, 3pi/2]$ and $[3pi/2, 2pi]$, is this correct? If so, how can this be done?
calculus analysis trigonometry power-series
calculus analysis trigonometry power-series
asked Nov 17 at 20:22
Surzilla
1238
asked Nov 17 at 20:22
Surzilla
1238
edited Nov 18 at 9:21
asked Nov 17 at 20:22
Surzilla
1238
asked Nov 17 at 20:22
Surzilla
1238
asked Nov 17 at 20:22
Surzilla
1238
1238
The reverse question is asked here: "Deriving the power series for cosine, using basic geometry". You may find my answer interesting.
– Blue
Nov 17 at 20:25
1
@Blue thank you, but I can't quite follow the proof in that answer, especially since the combinatorial argument is omitted. I appreciate the reference and will probably return to it when my math skills are stronger, but for now I'd appreciate a proof in the direction I asked.
– Surzilla
Nov 17 at 21:01
1
Possible duplicate: How to prove periodicity of $sin(x)$ and $cos(x)$ starting from the Taylor seires expansion?
– YiFan
Nov 18 at 10:44
add a comment |
The reverse question is asked here: "Deriving the power series for cosine, using basic geometry". You may find my answer interesting.
– Blue
Nov 17 at 20:25
1
@Blue thank you, but I can't quite follow the proof in that answer, especially since the combinatorial argument is omitted. I appreciate the reference and will probably return to it when my math skills are stronger, but for now I'd appreciate a proof in the direction I asked.
– Surzilla
Nov 17 at 21:01
1
Possible duplicate: How to prove periodicity of $sin(x)$ and $cos(x)$ starting from the Taylor seires expansion?
– YiFan
Nov 18 at 10:44
The reverse question is asked here: "Deriving the power series for cosine, using basic geometry". You may find my answer interesting.
– Blue
Nov 17 at 20:25
The reverse question is asked here: "Deriving the power series for cosine, using basic geometry". You may find my answer interesting.
– Blue
Nov 17 at 20:25
1
1
@Blue thank you, but I can't quite follow the proof in that answer, especially since the combinatorial argument is omitted. I appreciate the reference and will probably return to it when my math skills are stronger, but for now I'd appreciate a proof in the direction I asked.
– Surzilla
Nov 17 at 21:01
@Blue thank you, but I can't quite follow the proof in that answer, especially since the combinatorial argument is omitted. I appreciate the reference and will probably return to it when my math skills are stronger, but for now I'd appreciate a proof in the direction I asked.
– Surzilla
Nov 17 at 21:01
1
1
Possible duplicate: How to prove periodicity of $sin(x)$ and $cos(x)$ starting from the Taylor seires expansion?
– YiFan
Nov 18 at 10:44
Possible duplicate: How to prove periodicity of $sin(x)$ and $cos(x)$ starting from the Taylor seires expansion?
– YiFan
Nov 18 at 10:44
add a comment |
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The reverse question is asked here: "Deriving the power series for cosine, using basic geometry". You may find my answer interesting.
– Blue
Nov 17 at 20:25
1
@Blue thank you, but I can't quite follow the proof in that answer, especially since the combinatorial argument is omitted. I appreciate the reference and will probably return to it when my math skills are stronger, but for now I'd appreciate a proof in the direction I asked.
– Surzilla
Nov 17 at 21:01
1
Possible duplicate: How to prove periodicity of $sin(x)$ and $cos(x)$ starting from the Taylor seires expansion?
– YiFan
Nov 18 at 10:44