Evaluating a general solution to $5cos theta -12sin theta = 13$ using vectors [duplicate]
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This question already has an answer here:
Determining the general solution for the trigonometric equation $ 5cos(x)-12sin (x) = 13 $
5 answers
Given that
$$5cos theta -12sin theta = 13$$
I'm trying to evaluate a general solution for this equation. It appears I'll be using vector product.
My equation is equivalent to
$$langle (5,12), (costheta, sintheta)rangle = 13$$
which yields (by Cauch Schwarz Inequality)
$$|langle (5,12), (costheta, sintheta)rangle| le |(5,12)||(costheta, sintheta)| = 13$$
This is where I'm stuck.
Regards
trigonometry vectors vector-analysis
edited Sep 3 at 14:41
Blue
44.2k868141
asked Sep 3 at 14:25
Busi
32218
marked as duplicate by Michael Seifert, cansomeonehelpmeout, Arnaud D., Mark Bennet, user99914 Sep 3 at 16:02
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
 |Â
show 1 more comment
up vote
0
down vote
favorite
This question already has an answer here:
Determining the general solution for the trigonometric equation $ 5cos(x)-12sin (x) = 13 $
5 answers
Given that
$$5cos theta -12sin theta = 13$$
I'm trying to evaluate a general solution for this equation. It appears I'll be using vector product.
My equation is equivalent to
$$langle (5,12), (costheta, sintheta)rangle = 13$$
which yields (by Cauch Schwarz Inequality)
$$|langle (5,12), (costheta, sintheta)rangle| le |(5,12)||(costheta, sintheta)| = 13$$
This is where I'm stuck.
Regards
trigonometry vectors vector-analysis
edited Sep 3 at 14:41
Blue
44.2k868141
asked Sep 3 at 14:25
Busi
32218
marked as duplicate by Michael Seifert, cansomeonehelpmeout, Arnaud D., Mark Bennet, user99914 Sep 3 at 16:02
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
Do you know when the Cauchy-Schwarz inequality is actually an equality?
– Arnaud D.
Sep 3 at 14:30
@ArnaudD. I truly do not. I'll be very glad if you can show. Even thought I googled it, there were not any useful results. That's what I'm actually missing.
– Busi
Sep 3 at 14:30
Wait, you've pretty much asked this question before : math.stackexchange.com/questions/2903190/…. If you have some trouble with some of the answers there, it's better to comment there.
– Arnaud D.
Sep 3 at 14:33
@ArnaudD. Yes, that's because I did not get it properly.
– Busi
Sep 3 at 14:33
So have you computed the length of each of the vectors?
– Mark Bennet
Sep 3 at 14:44
 |Â
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
This question already has an answer here:
Determining the general solution for the trigonometric equation $ 5cos(x)-12sin (x) = 13 $
5 answers
Given that
$$5cos theta -12sin theta = 13$$
I'm trying to evaluate a general solution for this equation. It appears I'll be using vector product.
My equation is equivalent to
$$langle (5,12), (costheta, sintheta)rangle = 13$$
which yields (by Cauch Schwarz Inequality)
$$|langle (5,12), (costheta, sintheta)rangle| le |(5,12)||(costheta, sintheta)| = 13$$
This is where I'm stuck.
Regards
trigonometry vectors vector-analysis
edited Sep 3 at 14:41
Blue
44.2k868141
asked Sep 3 at 14:25
Busi
32218
This question already has an answer here:
Determining the general solution for the trigonometric equation $ 5cos(x)-12sin (x) = 13 $
5 answers
Given that
$$5cos theta -12sin theta = 13$$
I'm trying to evaluate a general solution for this equation. It appears I'll be using vector product.
My equation is equivalent to
$$langle (5,12), (costheta, sintheta)rangle = 13$$
which yields (by Cauch Schwarz Inequality)
$$|langle (5,12), (costheta, sintheta)rangle| le |(5,12)||(costheta, sintheta)| = 13$$
This is where I'm stuck.
Regards
This question already has an answer here:
Determining the general solution for the trigonometric equation $ 5cos(x)-12sin (x) = 13 $
5 answers
trigonometry vectors vector-analysis
trigonometry vectors vector-analysis
edited Sep 3 at 14:41
Blue
44.2k868141
asked Sep 3 at 14:25
Busi
32218
edited Sep 3 at 14:41
Blue
44.2k868141
asked Sep 3 at 14:25
Busi
32218
edited Sep 3 at 14:41
Blue
44.2k868141
edited Sep 3 at 14:41
Blue
44.2k868141
edited Sep 3 at 14:41
Blue
44.2k868141
44.2k868141
asked Sep 3 at 14:25
Busi
32218
asked Sep 3 at 14:25
Busi
32218
asked Sep 3 at 14:25
Busi
32218
32218
marked as duplicate by Michael Seifert, cansomeonehelpmeout, Arnaud D., Mark Bennet, user99914 Sep 3 at 16:02
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Michael Seifert, cansomeonehelpmeout, Arnaud D., Mark Bennet, user99914 Sep 3 at 16:02
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
Do you know when the Cauchy-Schwarz inequality is actually an equality?
– Arnaud D.
Sep 3 at 14:30
@ArnaudD. I truly do not. I'll be very glad if you can show. Even thought I googled it, there were not any useful results. That's what I'm actually missing.
– Busi
Sep 3 at 14:30
Wait, you've pretty much asked this question before : math.stackexchange.com/questions/2903190/…. If you have some trouble with some of the answers there, it's better to comment there.
– Arnaud D.
Sep 3 at 14:33
@ArnaudD. Yes, that's because I did not get it properly.
– Busi
Sep 3 at 14:33
So have you computed the length of each of the vectors?
– Mark Bennet
Sep 3 at 14:44
 |Â
show 1 more comment
Do you know when the Cauchy-Schwarz inequality is actually an equality?
– Arnaud D.
Sep 3 at 14:30
@ArnaudD. I truly do not. I'll be very glad if you can show. Even thought I googled it, there were not any useful results. That's what I'm actually missing.
– Busi
Sep 3 at 14:30
Wait, you've pretty much asked this question before : math.stackexchange.com/questions/2903190/…. If you have some trouble with some of the answers there, it's better to comment there.
– Arnaud D.
Sep 3 at 14:33
@ArnaudD. Yes, that's because I did not get it properly.
– Busi
Sep 3 at 14:33
So have you computed the length of each of the vectors?
– Mark Bennet
Sep 3 at 14:44
Do you know when the Cauchy-Schwarz inequality is actually an equality?
– Arnaud D.
Sep 3 at 14:30
Do you know when the Cauchy-Schwarz inequality is actually an equality?
– Arnaud D.
Sep 3 at 14:30
@ArnaudD. I truly do not. I'll be very glad if you can show. Even thought I googled it, there were not any useful results. That's what I'm actually missing.
– Busi
Sep 3 at 14:30
@ArnaudD. I truly do not. I'll be very glad if you can show. Even thought I googled it, there were not any useful results. That's what I'm actually missing.
– Busi
Sep 3 at 14:30
Wait, you've pretty much asked this question before : math.stackexchange.com/questions/2903190/…. If you have some trouble with some of the answers there, it's better to comment there.
– Arnaud D.
Sep 3 at 14:33
Wait, you've pretty much asked this question before : math.stackexchange.com/questions/2903190/…. If you have some trouble with some of the answers there, it's better to comment there.
– Arnaud D.
Sep 3 at 14:33
@ArnaudD. Yes, that's because I did not get it properly.
– Busi
Sep 3 at 14:33
@ArnaudD. Yes, that's because I did not get it properly.
– Busi
Sep 3 at 14:33
So have you computed the length of each of the vectors?
– Mark Bennet
Sep 3 at 14:44
So have you computed the length of each of the vectors?
– Mark Bennet
Sep 3 at 14:44
 |Â
show 1 more comment
2 Answers
2
active
oldest
votes
up vote
2
down vote
accepted
Recall that given two vectors in $mathbbR^2$ or $mathbbR^3$ $u$ and $v$ by dot product we have
$$ucdot v=|u||v|cos theta$$
and since $-1le cos theta le 1$ we have
$$-|u||v|le ucdot vle |u||v|iff |ucdot v|le |u||v|$$
Since $|cos theta|=1$ when $theta=0, pi$ the equality holds if and only if $u$ and $v$ are multiple vectors.
The result can be generalized for any dimension and it is known as Cauchy-Schwarz inequality.
answered Sep 3 at 14:39
gimusi
72.6k73888
1
I don't see how this answers the question. OP seems to know the Cauchy-Schwarz inequality.
– Arnaud D.
Sep 3 at 14:41
1
@ArnaudD. Are you joking? Did you read the aswer of the OP to you own question?: Question: "Do you know when the Cauchy-Schwarz inequality is actually an equality?" - Answer: "I truly do not. I'll be very glad if you can show. Even thought I googled it, there were not any useful results. That's what I'm actually missing".
– gimusi
Sep 3 at 14:43
But your answer did not say anything about when the inequality is an equality, at the time I commented.
– Arnaud D.
Sep 3 at 14:46
@ArnaudD. Please explain your comment or you have a very short memory or you are in bad faith!
– gimusi
Sep 3 at 14:46
@ArnaudD. I've answered after your question and after OP answer. I think you should apologize and pay attention to your behaviour before to accuse someone.
– gimusi
Sep 3 at 14:47
 |Â
show 3 more comments
up vote
0
down vote
Hint.
beginalign
5cos theta -12sin theta = 13
&implies -12sin theta = 13-5cos theta
\
&implies (-12sin theta )^2= (13-5cos theta)^2
\
&implies 144(1-cos^2 theta)= 169-130cos theta +cos^2theta
\
&implies 144-144cos^2 theta= 169-130cos theta +cos^2theta
\
&implies 145cos^2 theta-130costheta+25=0
endalign
answered Sep 3 at 14:38
MathOverview
8,35442962
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Recall that given two vectors in $mathbbR^2$ or $mathbbR^3$ $u$ and $v$ by dot product we have
$$ucdot v=|u||v|cos theta$$
and since $-1le cos theta le 1$ we have
$$-|u||v|le ucdot vle |u||v|iff |ucdot v|le |u||v|$$
Since $|cos theta|=1$ when $theta=0, pi$ the equality holds if and only if $u$ and $v$ are multiple vectors.
The result can be generalized for any dimension and it is known as Cauchy-Schwarz inequality.
answered Sep 3 at 14:39
gimusi
72.6k73888
1
I don't see how this answers the question. OP seems to know the Cauchy-Schwarz inequality.
– Arnaud D.
Sep 3 at 14:41
1
@ArnaudD. Are you joking? Did you read the aswer of the OP to you own question?: Question: "Do you know when the Cauchy-Schwarz inequality is actually an equality?" - Answer: "I truly do not. I'll be very glad if you can show. Even thought I googled it, there were not any useful results. That's what I'm actually missing".
– gimusi
Sep 3 at 14:43
But your answer did not say anything about when the inequality is an equality, at the time I commented.
– Arnaud D.
Sep 3 at 14:46
@ArnaudD. Please explain your comment or you have a very short memory or you are in bad faith!
– gimusi
Sep 3 at 14:46
@ArnaudD. I've answered after your question and after OP answer. I think you should apologize and pay attention to your behaviour before to accuse someone.
– gimusi
Sep 3 at 14:47
 |Â
show 3 more comments
up vote
2
down vote
accepted
Recall that given two vectors in $mathbbR^2$ or $mathbbR^3$ $u$ and $v$ by dot product we have
$$ucdot v=|u||v|cos theta$$
and since $-1le cos theta le 1$ we have
$$-|u||v|le ucdot vle |u||v|iff |ucdot v|le |u||v|$$
Since $|cos theta|=1$ when $theta=0, pi$ the equality holds if and only if $u$ and $v$ are multiple vectors.
The result can be generalized for any dimension and it is known as Cauchy-Schwarz inequality.
answered Sep 3 at 14:39
gimusi
72.6k73888
1
I don't see how this answers the question. OP seems to know the Cauchy-Schwarz inequality.
– Arnaud D.
Sep 3 at 14:41
1
@ArnaudD. Are you joking? Did you read the aswer of the OP to you own question?: Question: "Do you know when the Cauchy-Schwarz inequality is actually an equality?" - Answer: "I truly do not. I'll be very glad if you can show. Even thought I googled it, there were not any useful results. That's what I'm actually missing".
– gimusi
Sep 3 at 14:43
But your answer did not say anything about when the inequality is an equality, at the time I commented.
– Arnaud D.
Sep 3 at 14:46
@ArnaudD. Please explain your comment or you have a very short memory or you are in bad faith!
– gimusi
Sep 3 at 14:46
@ArnaudD. I've answered after your question and after OP answer. I think you should apologize and pay attention to your behaviour before to accuse someone.
– gimusi
Sep 3 at 14:47
 |Â
show 3 more comments
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Recall that given two vectors in $mathbbR^2$ or $mathbbR^3$ $u$ and $v$ by dot product we have
$$ucdot v=|u||v|cos theta$$
and since $-1le cos theta le 1$ we have
$$-|u||v|le ucdot vle |u||v|iff |ucdot v|le |u||v|$$
Since $|cos theta|=1$ when $theta=0, pi$ the equality holds if and only if $u$ and $v$ are multiple vectors.
The result can be generalized for any dimension and it is known as Cauchy-Schwarz inequality.
answered Sep 3 at 14:39
gimusi
72.6k73888
Recall that given two vectors in $mathbbR^2$ or $mathbbR^3$ $u$ and $v$ by dot product we have
$$ucdot v=|u||v|cos theta$$
and since $-1le cos theta le 1$ we have
$$-|u||v|le ucdot vle |u||v|iff |ucdot v|le |u||v|$$
Since $|cos theta|=1$ when $theta=0, pi$ the equality holds if and only if $u$ and $v$ are multiple vectors.
The result can be generalized for any dimension and it is known as Cauchy-Schwarz inequality.
answered Sep 3 at 14:39
gimusi
72.6k73888
edited Sep 3 at 14:42
answered Sep 3 at 14:39
gimusi
72.6k73888
answered Sep 3 at 14:39
gimusi
72.6k73888
answered Sep 3 at 14:39
gimusi
72.6k73888
72.6k73888
1
I don't see how this answers the question. OP seems to know the Cauchy-Schwarz inequality.
– Arnaud D.
Sep 3 at 14:41
1
@ArnaudD. Are you joking? Did you read the aswer of the OP to you own question?: Question: "Do you know when the Cauchy-Schwarz inequality is actually an equality?" - Answer: "I truly do not. I'll be very glad if you can show. Even thought I googled it, there were not any useful results. That's what I'm actually missing".
– gimusi
Sep 3 at 14:43
But your answer did not say anything about when the inequality is an equality, at the time I commented.
– Arnaud D.
Sep 3 at 14:46
@ArnaudD. Please explain your comment or you have a very short memory or you are in bad faith!
– gimusi
Sep 3 at 14:46
@ArnaudD. I've answered after your question and after OP answer. I think you should apologize and pay attention to your behaviour before to accuse someone.
– gimusi
Sep 3 at 14:47
 |Â
show 3 more comments
1
I don't see how this answers the question. OP seems to know the Cauchy-Schwarz inequality.
– Arnaud D.
Sep 3 at 14:41
1
@ArnaudD. Are you joking? Did you read the aswer of the OP to you own question?: Question: "Do you know when the Cauchy-Schwarz inequality is actually an equality?" - Answer: "I truly do not. I'll be very glad if you can show. Even thought I googled it, there were not any useful results. That's what I'm actually missing".
– gimusi
Sep 3 at 14:43
But your answer did not say anything about when the inequality is an equality, at the time I commented.
– Arnaud D.
Sep 3 at 14:46
@ArnaudD. Please explain your comment or you have a very short memory or you are in bad faith!
– gimusi
Sep 3 at 14:46
@ArnaudD. I've answered after your question and after OP answer. I think you should apologize and pay attention to your behaviour before to accuse someone.
– gimusi
Sep 3 at 14:47
1
1
I don't see how this answers the question. OP seems to know the Cauchy-Schwarz inequality.
– Arnaud D.
Sep 3 at 14:41
I don't see how this answers the question. OP seems to know the Cauchy-Schwarz inequality.
– Arnaud D.
Sep 3 at 14:41
1
1
@ArnaudD. Are you joking? Did you read the aswer of the OP to you own question?: Question: "Do you know when the Cauchy-Schwarz inequality is actually an equality?" - Answer: "I truly do not. I'll be very glad if you can show. Even thought I googled it, there were not any useful results. That's what I'm actually missing".
– gimusi
Sep 3 at 14:43
@ArnaudD. Are you joking? Did you read the aswer of the OP to you own question?: Question: "Do you know when the Cauchy-Schwarz inequality is actually an equality?" - Answer: "I truly do not. I'll be very glad if you can show. Even thought I googled it, there were not any useful results. That's what I'm actually missing".
– gimusi
Sep 3 at 14:43
But your answer did not say anything about when the inequality is an equality, at the time I commented.
– Arnaud D.
Sep 3 at 14:46
But your answer did not say anything about when the inequality is an equality, at the time I commented.
– Arnaud D.
Sep 3 at 14:46
@ArnaudD. Please explain your comment or you have a very short memory or you are in bad faith!
– gimusi
Sep 3 at 14:46
@ArnaudD. Please explain your comment or you have a very short memory or you are in bad faith!
– gimusi
Sep 3 at 14:46
@ArnaudD. I've answered after your question and after OP answer. I think you should apologize and pay attention to your behaviour before to accuse someone.
– gimusi
Sep 3 at 14:47
@ArnaudD. I've answered after your question and after OP answer. I think you should apologize and pay attention to your behaviour before to accuse someone.
– gimusi
Sep 3 at 14:47
 |Â
show 3 more comments
up vote
0
down vote
Hint.
beginalign
5cos theta -12sin theta = 13
&implies -12sin theta = 13-5cos theta
\
&implies (-12sin theta )^2= (13-5cos theta)^2
\
&implies 144(1-cos^2 theta)= 169-130cos theta +cos^2theta
\
&implies 144-144cos^2 theta= 169-130cos theta +cos^2theta
\
&implies 145cos^2 theta-130costheta+25=0
endalign
answered Sep 3 at 14:38
MathOverview
8,35442962
add a comment |Â
up vote
0
down vote
Hint.
beginalign
5cos theta -12sin theta = 13
&implies -12sin theta = 13-5cos theta
\
&implies (-12sin theta )^2= (13-5cos theta)^2
\
&implies 144(1-cos^2 theta)= 169-130cos theta +cos^2theta
\
&implies 144-144cos^2 theta= 169-130cos theta +cos^2theta
\
&implies 145cos^2 theta-130costheta+25=0
endalign
answered Sep 3 at 14:38
MathOverview
8,35442962
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Hint.
beginalign
5cos theta -12sin theta = 13
&implies -12sin theta = 13-5cos theta
\
&implies (-12sin theta )^2= (13-5cos theta)^2
\
&implies 144(1-cos^2 theta)= 169-130cos theta +cos^2theta
\
&implies 144-144cos^2 theta= 169-130cos theta +cos^2theta
\
&implies 145cos^2 theta-130costheta+25=0
endalign
answered Sep 3 at 14:38
MathOverview
8,35442962
Hint.
beginalign
5cos theta -12sin theta = 13
&implies -12sin theta = 13-5cos theta
\
&implies (-12sin theta )^2= (13-5cos theta)^2
\
&implies 144(1-cos^2 theta)= 169-130cos theta +cos^2theta
\
&implies 144-144cos^2 theta= 169-130cos theta +cos^2theta
\
&implies 145cos^2 theta-130costheta+25=0
endalign
answered Sep 3 at 14:38
MathOverview
8,35442962
answered Sep 3 at 14:38
MathOverview
8,35442962
answered Sep 3 at 14:38
MathOverview
8,35442962
answered Sep 3 at 14:38
MathOverview
8,35442962
8,35442962
add a comment |Â
add a comment |Â
Do you know when the Cauchy-Schwarz inequality is actually an equality?
– Arnaud D.
Sep 3 at 14:30
@ArnaudD. I truly do not. I'll be very glad if you can show. Even thought I googled it, there were not any useful results. That's what I'm actually missing.
– Busi
Sep 3 at 14:30
Wait, you've pretty much asked this question before : math.stackexchange.com/questions/2903190/…. If you have some trouble with some of the answers there, it's better to comment there.
– Arnaud D.
Sep 3 at 14:33
@ArnaudD. Yes, that's because I did not get it properly.
– Busi
Sep 3 at 14:33
So have you computed the length of each of the vectors?
– Mark Bennet
Sep 3 at 14:44