Use $cos 5theta$ to find the roots of $x(16x^4 - 20x^2 + 5) = 0$
$begingroup$
I used $cos(3theta + 2theta)$ to prove the first part, but I don't know how to the $2$nd part.
Show that $cos 5theta=16cos^5theta-20cos^3theta+5costheta,$ and hence show that
$$text{the roots of }x(16x^4 - 20x^2 + 5) text{ are: } 0,cosfrac{pi}{10}, cosfrac{3pi}{10},cosfrac{7pi}{10}, cosfrac{9pi}{10}$$
trigonometry complex-numbers
edited Nov 25 '18 at 17:28
Lorenzo B.
1,8402520
asked Nov 25 '18 at 12:40
VanessaVanessa
727
$endgroup$
add a comment |
$begingroup$
I used $cos(3theta + 2theta)$ to prove the first part, but I don't know how to the $2$nd part.
Show that $cos 5theta=16cos^5theta-20cos^3theta+5costheta,$ and hence show that
$$text{the roots of }x(16x^4 - 20x^2 + 5) text{ are: } 0,cosfrac{pi}{10}, cosfrac{3pi}{10},cosfrac{7pi}{10}, cosfrac{9pi}{10}$$
trigonometry complex-numbers
edited Nov 25 '18 at 17:28
Lorenzo B.
1,8402520
asked Nov 25 '18 at 12:40
VanessaVanessa
727
$endgroup$
add a comment |
$begingroup$
I used $cos(3theta + 2theta)$ to prove the first part, but I don't know how to the $2$nd part.
Show that $cos 5theta=16cos^5theta-20cos^3theta+5costheta,$ and hence show that
$$text{the roots of }x(16x^4 - 20x^2 + 5) text{ are: } 0,cosfrac{pi}{10}, cosfrac{3pi}{10},cosfrac{7pi}{10}, cosfrac{9pi}{10}$$
trigonometry complex-numbers
edited Nov 25 '18 at 17:28
Lorenzo B.
1,8402520
asked Nov 25 '18 at 12:40
VanessaVanessa
727
$endgroup$
I used $cos(3theta + 2theta)$ to prove the first part, but I don't know how to the $2$nd part.
Show that $cos 5theta=16cos^5theta-20cos^3theta+5costheta,$ and hence show that
$$text{the roots of }x(16x^4 - 20x^2 + 5) text{ are: } 0,cosfrac{pi}{10}, cosfrac{3pi}{10},cosfrac{7pi}{10}, cosfrac{9pi}{10}$$
trigonometry complex-numbers
trigonometry complex-numbers
edited Nov 25 '18 at 17:28
Lorenzo B.
1,8402520
asked Nov 25 '18 at 12:40
VanessaVanessa
727
edited Nov 25 '18 at 17:28
Lorenzo B.
1,8402520
asked Nov 25 '18 at 12:40
VanessaVanessa
727
edited Nov 25 '18 at 17:28
Lorenzo B.
1,8402520
edited Nov 25 '18 at 17:28
Lorenzo B.
1,8402520
edited Nov 25 '18 at 17:28
Lorenzo B.
1,8402520
1,8402520
asked Nov 25 '18 at 12:40
VanessaVanessa
727
asked Nov 25 '18 at 12:40
VanessaVanessa
727
asked Nov 25 '18 at 12:40
VanessaVanessa
727
727
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Here the value of cosine will repeat after point.
Final values wil be only 5
answered Nov 25 '18 at 16:56
kapil pundirkapil pundir
574
$endgroup$
1
$begingroup$
Please use MathJax for writing answers. Because in MSE app its difficult to use image for some smartphones.
$endgroup$
– jayant98
Nov 25 '18 at 17:19
$begingroup$
I am using android phone, it is not showing option for Mathjax while writing answer
$endgroup$
– kapil pundir
Nov 25 '18 at 17:32
$begingroup$
math.meta.stackexchange.com/questions/5020/…
$endgroup$
– jayant98
Nov 25 '18 at 18:13
$begingroup$
Thankyou very much
$endgroup$
– kapil pundir
Nov 26 '18 at 15:59
add a comment |
$begingroup$
The equation $$;x(16x^4 - 20x^2 + 5) = 0; tag 1$$ can have $5$ real solutions (incl. multiplicity).
From $;cos 5theta = 16cos^5 theta - 20cos^3 theta + 5;$ we deduce that we will look for solutions of $(1)$ in the form of $cos theta;$ where $;cos 5theta = 0.$
$cos 5theta = 0$ holds for $10$ different values of $;theta in (-pi,pi];$ obtained when solving $$cos 5theta =frac pi2 +2kpiquad text{or} quad cos 5theta = -frac pi2 + 2kpi,; k=0,1,dots,4.$$
These are $$frac{pi}{10},; frac{5pi}{10},; frac{9pi}{10},;frac{13pi}{10},;frac{18pi}{10};text{and};-frac{pi}{10},;frac{3pi}{10},;frac{7pi}{10},;frac{11pi}{10},;frac{15pi}{10},;$$
between them two and two have equal cosines (can you check which are the pairs?).
The solutions are those listed in the question.
edited Nov 25 '18 at 17:07
amWhy
1
answered Nov 25 '18 at 15:21
user376343user376343
3,3582826
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3012786%2fuse-cos-5-theta-to-find-the-roots-of-x16x4-20x2-5-0%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Here the value of cosine will repeat after point.
Final values wil be only 5
answered Nov 25 '18 at 16:56
kapil pundirkapil pundir
574
$endgroup$
1
$begingroup$
Please use MathJax for writing answers. Because in MSE app its difficult to use image for some smartphones.
$endgroup$
– jayant98
Nov 25 '18 at 17:19
$begingroup$
I am using android phone, it is not showing option for Mathjax while writing answer
$endgroup$
– kapil pundir
Nov 25 '18 at 17:32
$begingroup$
math.meta.stackexchange.com/questions/5020/…
$endgroup$
– jayant98
Nov 25 '18 at 18:13
$begingroup$
Thankyou very much
$endgroup$
– kapil pundir
Nov 26 '18 at 15:59
add a comment |
$begingroup$
Here the value of cosine will repeat after point.
Final values wil be only 5
answered Nov 25 '18 at 16:56
kapil pundirkapil pundir
574
$endgroup$
1
$begingroup$
Please use MathJax for writing answers. Because in MSE app its difficult to use image for some smartphones.
$endgroup$
– jayant98
Nov 25 '18 at 17:19
$begingroup$
I am using android phone, it is not showing option for Mathjax while writing answer
$endgroup$
– kapil pundir
Nov 25 '18 at 17:32
$begingroup$
math.meta.stackexchange.com/questions/5020/…
$endgroup$
– jayant98
Nov 25 '18 at 18:13
$begingroup$
Thankyou very much
$endgroup$
– kapil pundir
Nov 26 '18 at 15:59
add a comment |
$begingroup$
Here the value of cosine will repeat after point.
Final values wil be only 5
answered Nov 25 '18 at 16:56
kapil pundirkapil pundir
574
$endgroup$
Here the value of cosine will repeat after point.
Final values wil be only 5
answered Nov 25 '18 at 16:56
kapil pundirkapil pundir
574
answered Nov 25 '18 at 16:56
kapil pundirkapil pundir
574
answered Nov 25 '18 at 16:56
kapil pundirkapil pundir
574
answered Nov 25 '18 at 16:56
kapil pundirkapil pundir
574
574
1
$begingroup$
Please use MathJax for writing answers. Because in MSE app its difficult to use image for some smartphones.
$endgroup$
– jayant98
Nov 25 '18 at 17:19
$begingroup$
I am using android phone, it is not showing option for Mathjax while writing answer
$endgroup$
– kapil pundir
Nov 25 '18 at 17:32
$begingroup$
math.meta.stackexchange.com/questions/5020/…
$endgroup$
– jayant98
Nov 25 '18 at 18:13
$begingroup$
Thankyou very much
$endgroup$
– kapil pundir
Nov 26 '18 at 15:59
add a comment |
1
$begingroup$
Please use MathJax for writing answers. Because in MSE app its difficult to use image for some smartphones.
$endgroup$
– jayant98
Nov 25 '18 at 17:19
$begingroup$
I am using android phone, it is not showing option for Mathjax while writing answer
$endgroup$
– kapil pundir
Nov 25 '18 at 17:32
$begingroup$
math.meta.stackexchange.com/questions/5020/…
$endgroup$
– jayant98
Nov 25 '18 at 18:13
$begingroup$
Thankyou very much
$endgroup$
– kapil pundir
Nov 26 '18 at 15:59
1
1
$begingroup$
Please use MathJax for writing answers. Because in MSE app its difficult to use image for some smartphones.
$endgroup$
– jayant98
Nov 25 '18 at 17:19
$begingroup$
Please use MathJax for writing answers. Because in MSE app its difficult to use image for some smartphones.
$endgroup$
– jayant98
Nov 25 '18 at 17:19
$begingroup$
I am using android phone, it is not showing option for Mathjax while writing answer
$endgroup$
– kapil pundir
Nov 25 '18 at 17:32
$begingroup$
I am using android phone, it is not showing option for Mathjax while writing answer
$endgroup$
– kapil pundir
Nov 25 '18 at 17:32
$begingroup$
math.meta.stackexchange.com/questions/5020/…
$endgroup$
– jayant98
Nov 25 '18 at 18:13
$begingroup$
math.meta.stackexchange.com/questions/5020/…
$endgroup$
– jayant98
Nov 25 '18 at 18:13
$begingroup$
Thankyou very much
$endgroup$
– kapil pundir
Nov 26 '18 at 15:59
$begingroup$
Thankyou very much
$endgroup$
– kapil pundir
Nov 26 '18 at 15:59
add a comment |
$begingroup$
The equation $$;x(16x^4 - 20x^2 + 5) = 0; tag 1$$ can have $5$ real solutions (incl. multiplicity).
From $;cos 5theta = 16cos^5 theta - 20cos^3 theta + 5;$ we deduce that we will look for solutions of $(1)$ in the form of $cos theta;$ where $;cos 5theta = 0.$
$cos 5theta = 0$ holds for $10$ different values of $;theta in (-pi,pi];$ obtained when solving $$cos 5theta =frac pi2 +2kpiquad text{or} quad cos 5theta = -frac pi2 + 2kpi,; k=0,1,dots,4.$$
These are $$frac{pi}{10},; frac{5pi}{10},; frac{9pi}{10},;frac{13pi}{10},;frac{18pi}{10};text{and};-frac{pi}{10},;frac{3pi}{10},;frac{7pi}{10},;frac{11pi}{10},;frac{15pi}{10},;$$
between them two and two have equal cosines (can you check which are the pairs?).
The solutions are those listed in the question.
edited Nov 25 '18 at 17:07
amWhy
1
answered Nov 25 '18 at 15:21
user376343user376343
3,3582826
$endgroup$
add a comment |
$begingroup$
The equation $$;x(16x^4 - 20x^2 + 5) = 0; tag 1$$ can have $5$ real solutions (incl. multiplicity).
From $;cos 5theta = 16cos^5 theta - 20cos^3 theta + 5;$ we deduce that we will look for solutions of $(1)$ in the form of $cos theta;$ where $;cos 5theta = 0.$
$cos 5theta = 0$ holds for $10$ different values of $;theta in (-pi,pi];$ obtained when solving $$cos 5theta =frac pi2 +2kpiquad text{or} quad cos 5theta = -frac pi2 + 2kpi,; k=0,1,dots,4.$$
These are $$frac{pi}{10},; frac{5pi}{10},; frac{9pi}{10},;frac{13pi}{10},;frac{18pi}{10};text{and};-frac{pi}{10},;frac{3pi}{10},;frac{7pi}{10},;frac{11pi}{10},;frac{15pi}{10},;$$
between them two and two have equal cosines (can you check which are the pairs?).
The solutions are those listed in the question.
edited Nov 25 '18 at 17:07
amWhy
1
answered Nov 25 '18 at 15:21
user376343user376343
3,3582826
$endgroup$
add a comment |
$begingroup$
The equation $$;x(16x^4 - 20x^2 + 5) = 0; tag 1$$ can have $5$ real solutions (incl. multiplicity).
From $;cos 5theta = 16cos^5 theta - 20cos^3 theta + 5;$ we deduce that we will look for solutions of $(1)$ in the form of $cos theta;$ where $;cos 5theta = 0.$
$cos 5theta = 0$ holds for $10$ different values of $;theta in (-pi,pi];$ obtained when solving $$cos 5theta =frac pi2 +2kpiquad text{or} quad cos 5theta = -frac pi2 + 2kpi,; k=0,1,dots,4.$$
These are $$frac{pi}{10},; frac{5pi}{10},; frac{9pi}{10},;frac{13pi}{10},;frac{18pi}{10};text{and};-frac{pi}{10},;frac{3pi}{10},;frac{7pi}{10},;frac{11pi}{10},;frac{15pi}{10},;$$
between them two and two have equal cosines (can you check which are the pairs?).
The solutions are those listed in the question.
edited Nov 25 '18 at 17:07
amWhy
1
answered Nov 25 '18 at 15:21
user376343user376343
3,3582826
$endgroup$
The equation $$;x(16x^4 - 20x^2 + 5) = 0; tag 1$$ can have $5$ real solutions (incl. multiplicity).
From $;cos 5theta = 16cos^5 theta - 20cos^3 theta + 5;$ we deduce that we will look for solutions of $(1)$ in the form of $cos theta;$ where $;cos 5theta = 0.$
$cos 5theta = 0$ holds for $10$ different values of $;theta in (-pi,pi];$ obtained when solving $$cos 5theta =frac pi2 +2kpiquad text{or} quad cos 5theta = -frac pi2 + 2kpi,; k=0,1,dots,4.$$
These are $$frac{pi}{10},; frac{5pi}{10},; frac{9pi}{10},;frac{13pi}{10},;frac{18pi}{10};text{and};-frac{pi}{10},;frac{3pi}{10},;frac{7pi}{10},;frac{11pi}{10},;frac{15pi}{10},;$$
between them two and two have equal cosines (can you check which are the pairs?).
The solutions are those listed in the question.
edited Nov 25 '18 at 17:07
amWhy
1
answered Nov 25 '18 at 15:21
user376343user376343
3,3582826
edited Nov 25 '18 at 17:07
amWhy
1
edited Nov 25 '18 at 17:07
amWhy
1
edited Nov 25 '18 at 17:07
amWhy
1
1
answered Nov 25 '18 at 15:21
user376343user376343
3,3582826
answered Nov 25 '18 at 15:21
user376343user376343
3,3582826
answered Nov 25 '18 at 15:21
user376343user376343
3,3582826
3,3582826
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3012786%2fuse-cos-5-theta-to-find-the-roots-of-x16x4-20x2-5-0%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
