Fdtxjr

What does the word "Atten" mean?

How to check is there any negative term in a large list?

Is this apparent Class Action settlement a spam message?

Purchasing a ticket for someone else in another country?

Is there a problem with hiding "forgot password" until it's needed?

How can I get through very long and very dry, but also very useful technical documents when learning a new tool?

How can I kill an app using Terminal?

Opposite of a diet

Would this custom Sorcerer variant that can only learn any verbal-component-only spell be unbalanced?

How easy is it to start Magic from scratch?

How do I go from 300 unfinished/half written blog posts, to published posts?

Pole-zeros of a real-valued causal FIR system

Escape a backup date in a file name

What is paid subscription needed for in Mortal Kombat 11?

What is the best translation for "slot" in the context of multiplayer video games?

Why escape if the_content isnt?

Hostile work environment after whistle-blowing on coworker and our boss. What do I do?

Is there a good way to store credentials outside of a password manager?

Proof of work - lottery approach

Lay out the Carpet

Sequence of Tenses: Translating the subjunctive

Avoiding estate tax by giving multiple gifts

Unreliable Magic - Is it worth it?

What is the opposite of 'gravitas'?

Understanding Trig equations

Trig identity proof helpTrig equation help pleaseConfused by textbook solution to trig problema trig questionGeneral solutions for trigonometry equationsSimple Trig Equations - Why is it Wrong to Cancel Trig Terms?Extracting Like Variable Nested in Trig FunctionsIf $tan t = frac14$ and the terminal point for t is in Quadrant III, find $sec t + cot t$.What is wrong with this way of solving trig equations?If $sinx=t, quad xin(frac3pi2,2pi),$ what is $tanx?$

0

$begingroup$

I understand why $sin(x)=fracsqrt32$ has two answers, $fracpi3$ and $frac2pi3$ but I don't understand why $tan(x)=1$ only has one solution (according to my book and other places I've looked online) $fracpi4$. Why isn't $frac5pi4$ also a solution given that tan is positive in Q1 and Q3 and at $frac5pi4$ both $sin$ and $cos$ are $-fracsqrt22$

Note that the interval is not given.

trigonometry

share|cite|improve this question

edited Mar 17 at 21:36

dstarh

asked Mar 17 at 21:23

dstarhdstarh

1426

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  $sin(x) = sqrt3/2$ has infinitely many solutions, as does $tan(x) = 0$. That being said, I don't think that $pi/3$ solves $tan(x) = 0$. Nor do I agree with your assertion that $sin(4pi/3) = cos(4pi/3) = -sqrt2/2$.
  $endgroup$
  – Xander Henderson
  Mar 17 at 21:27
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry meant $frac5pi4$ editing
  $endgroup$
  – dstarh
  Mar 17 at 21:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you mean $tan(x) = 1$? Regardless, both equations have infinitely many solutions as @XanderHenderson mentioned.
  $endgroup$
  – Klaus
  Mar 17 at 21:30
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You ask "why isn't $frac5pi4$ also a solution," yet $x=frac5pi4$ is indeed a solution: $tan(frac5pi4)=1$
  $endgroup$
  – Noble Mushtak
  Mar 17 at 21:33
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, edited to fix. So in this case since there is no interval given it's assumed that it's just $fracpi4 + pi*n$ ?
  $endgroup$
  – dstarh
  Mar 17 at 21:33
  
  

  
  
  
  

 | 
show 1 more comment

0

$begingroup$

I understand why $sin(x)=fracsqrt32$ has two answers, $fracpi3$ and $frac2pi3$ but I don't understand why $tan(x)=1$ only has one solution (according to my book and other places I've looked online) $fracpi4$. Why isn't $frac5pi4$ also a solution given that tan is positive in Q1 and Q3 and at $frac5pi4$ both $sin$ and $cos$ are $-fracsqrt22$

Note that the interval is not given.

trigonometry

share|cite|improve this question

edited Mar 17 at 21:36

dstarh

asked Mar 17 at 21:23

dstarhdstarh

1426

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  $sin(x) = sqrt3/2$ has infinitely many solutions, as does $tan(x) = 0$. That being said, I don't think that $pi/3$ solves $tan(x) = 0$. Nor do I agree with your assertion that $sin(4pi/3) = cos(4pi/3) = -sqrt2/2$.
  $endgroup$
  – Xander Henderson
  Mar 17 at 21:27
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry meant $frac5pi4$ editing
  $endgroup$
  – dstarh
  Mar 17 at 21:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you mean $tan(x) = 1$? Regardless, both equations have infinitely many solutions as @XanderHenderson mentioned.
  $endgroup$
  – Klaus
  Mar 17 at 21:30
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You ask "why isn't $frac5pi4$ also a solution," yet $x=frac5pi4$ is indeed a solution: $tan(frac5pi4)=1$
  $endgroup$
  – Noble Mushtak
  Mar 17 at 21:33
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, edited to fix. So in this case since there is no interval given it's assumed that it's just $fracpi4 + pi*n$ ?
  $endgroup$
  – dstarh
  Mar 17 at 21:33
  
  

  
  
  
  

 | 
show 1 more comment

$begingroup$

I understand why $sin(x)=fracsqrt32$ has two answers, $fracpi3$ and $frac2pi3$ but I don't understand why $tan(x)=1$ only has one solution (according to my book and other places I've looked online) $fracpi4$. Why isn't $frac5pi4$ also a solution given that tan is positive in Q1 and Q3 and at $frac5pi4$ both $sin$ and $cos$ are $-fracsqrt22$

Note that the interval is not given.

trigonometry

share|cite|improve this question

edited Mar 17 at 21:36

dstarh

asked Mar 17 at 21:23

dstarhdstarh

1426

$endgroup$

share|cite|improve this question

edited Mar 17 at 21:36

dstarh

asked Mar 17 at 21:23

dstarhdstarh

1426

share|cite|improve this question

edited Mar 17 at 21:36

dstarh

asked Mar 17 at 21:23

dstarhdstarh

1426

asked Mar 17 at 21:23

dstarhdstarh

1426

asked Mar 17 at 21:23

dstarhdstarh

1426

1 Answer
1

active

oldest

votes

1

$begingroup$

If the interval is not given, you would be wise to assume it's from minus infinity to positive infinity.

Consequently you should give the general solution.

So,
$$sin(x)=fracsqrt32$$
has solutions,
$$fracpi3+2pi k$$
or
$$frac2pi3+2pi k$$
for integer values of $k$

Shall I leave you to do similar for the other equations in your question ?

Top Tip : Drawing a graph of the appropriate trigonometric function is helpful.

share|cite|improve this answer

answered Mar 17 at 21:33

Martin HansenMartin Hansen

707114

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok since there are 2 general solutions to sin(x) then you list both but since it's only one technical solution for tan(x) you would only list the one? That makes more sense. Thanks
  $endgroup$
  – dstarh
  Mar 17 at 21:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If the interval IS given for ex from $0 < x < 2pi$ then would both solutions be valid for the tan equation?
  $endgroup$
  – dstarh
  Mar 17 at 21:37
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The period of the $tan(x)$ graph is just $pi$ radians so you can typically compact it's general solutions more than for general solutions from the $sin(x)$ and $cos(x)$ graphs with their period of $2pi$
  $endgroup$
  – Martin Hansen
  Mar 17 at 21:38
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ahh that makes sense. For whatever reason my professor skipped graphing of tan and just focused on sin/cos
  $endgroup$
  – dstarh
  Mar 17 at 21:39
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Just to be clear : $tan(x)=1$ has the two solutions $fracpi4$ and $frac5pi4$ in the interval $0<x<2pi$
  $endgroup$
  – Martin Hansen
  Mar 17 at 21:41
  
  

  
  
  
  

 | 
show 2 more comments

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
);



);

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3152106%2funderstanding-trig-equations%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

1

$begingroup$

If the interval is not given, you would be wise to assume it's from minus infinity to positive infinity.

Consequently you should give the general solution.

So,
$$sin(x)=fracsqrt32$$
has solutions,
$$fracpi3+2pi k$$
or
$$frac2pi3+2pi k$$
for integer values of $k$

Shall I leave you to do similar for the other equations in your question ?

Top Tip : Drawing a graph of the appropriate trigonometric function is helpful.

share|cite|improve this answer

answered Mar 17 at 21:33

Martin HansenMartin Hansen

707114

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok since there are 2 general solutions to sin(x) then you list both but since it's only one technical solution for tan(x) you would only list the one? That makes more sense. Thanks
  $endgroup$
  – dstarh
  Mar 17 at 21:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If the interval IS given for ex from $0 < x < 2pi$ then would both solutions be valid for the tan equation?
  $endgroup$
  – dstarh
  Mar 17 at 21:37
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The period of the $tan(x)$ graph is just $pi$ radians so you can typically compact it's general solutions more than for general solutions from the $sin(x)$ and $cos(x)$ graphs with their period of $2pi$
  $endgroup$
  – Martin Hansen
  Mar 17 at 21:38
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ahh that makes sense. For whatever reason my professor skipped graphing of tan and just focused on sin/cos
  $endgroup$
  – dstarh
  Mar 17 at 21:39
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Just to be clear : $tan(x)=1$ has the two solutions $fracpi4$ and $frac5pi4$ in the interval $0<x<2pi$
  $endgroup$
  – Martin Hansen
  Mar 17 at 21:41
  
  

  
  
  
  

 | 
show 2 more comments

1

$begingroup$

If the interval is not given, you would be wise to assume it's from minus infinity to positive infinity.

Consequently you should give the general solution.

So,
$$sin(x)=fracsqrt32$$
has solutions,
$$fracpi3+2pi k$$
or
$$frac2pi3+2pi k$$
for integer values of $k$

Shall I leave you to do similar for the other equations in your question ?

Top Tip : Drawing a graph of the appropriate trigonometric function is helpful.

share|cite|improve this answer

answered Mar 17 at 21:33

Martin HansenMartin Hansen

707114

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok since there are 2 general solutions to sin(x) then you list both but since it's only one technical solution for tan(x) you would only list the one? That makes more sense. Thanks
  $endgroup$
  – dstarh
  Mar 17 at 21:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If the interval IS given for ex from $0 < x < 2pi$ then would both solutions be valid for the tan equation?
  $endgroup$
  – dstarh
  Mar 17 at 21:37
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The period of the $tan(x)$ graph is just $pi$ radians so you can typically compact it's general solutions more than for general solutions from the $sin(x)$ and $cos(x)$ graphs with their period of $2pi$
  $endgroup$
  – Martin Hansen
  Mar 17 at 21:38
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ahh that makes sense. For whatever reason my professor skipped graphing of tan and just focused on sin/cos
  $endgroup$
  – dstarh
  Mar 17 at 21:39
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Just to be clear : $tan(x)=1$ has the two solutions $fracpi4$ and $frac5pi4$ in the interval $0<x<2pi$
  $endgroup$
  – Martin Hansen
  Mar 17 at 21:41
  
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

If the interval is not given, you would be wise to assume it's from minus infinity to positive infinity.

Consequently you should give the general solution.

So,
$$sin(x)=fracsqrt32$$
has solutions,
$$fracpi3+2pi k$$
or
$$frac2pi3+2pi k$$
for integer values of $k$

Shall I leave you to do similar for the other equations in your question ?

Top Tip : Drawing a graph of the appropriate trigonometric function is helpful.

share|cite|improve this answer

answered Mar 17 at 21:33

Martin HansenMartin Hansen

707114

$endgroup$

share|cite|improve this answer

answered Mar 17 at 21:33

Martin HansenMartin Hansen

707114

answered Mar 17 at 21:33

Martin HansenMartin Hansen

707114

answered Mar 17 at 21:33

Martin HansenMartin Hansen

707114