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On When And When Not To Divide By $tan(theta)$
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The textbook I'm using has this problem involving trig equations which goes $tan(x) sin^2(x)=2 tan(x)$; we're trying to solve for x. The textbook advises not to divide this problem by $tan(x)$ (in other words not to go $tan(x)/tan(x) sin^2(x)=2(tan(x)/tan(x))$.
I don't understand why that won't be a good idea. After all, when we try to prove trig identities (I can't remember which one), don't we also do the same thing, dividing out things like $tan(x)$ and $sin(x)$ in order to simplify it? Or at least in trig problems where we try to prove 2 things are equal? So why can't we do that right here?
I'm guessing it's because by dividing out $tan(x)$ in a problem where you are supposed to find the solution of $x$, you are dividing out a potential solution/root to the problem. But then, why can you divide out things like $tan(x)$ in a problem where we try to prove 2 things are equal, or when trying to prove a trig identity? Isn't that dividing out a potential trig relationship?
algebra-precalculus trigonometry
edited Jul 27 at 0:43
bjcolby15
7921616
asked Jul 26 at 13:44
Ethan Chan
608322
 |Â
show 4 more comments
up vote
0
down vote
favorite
The textbook I'm using has this problem involving trig equations which goes $tan(x) sin^2(x)=2 tan(x)$; we're trying to solve for x. The textbook advises not to divide this problem by $tan(x)$ (in other words not to go $tan(x)/tan(x) sin^2(x)=2(tan(x)/tan(x))$.
I don't understand why that won't be a good idea. After all, when we try to prove trig identities (I can't remember which one), don't we also do the same thing, dividing out things like $tan(x)$ and $sin(x)$ in order to simplify it? Or at least in trig problems where we try to prove 2 things are equal? So why can't we do that right here?
I'm guessing it's because by dividing out $tan(x)$ in a problem where you are supposed to find the solution of $x$, you are dividing out a potential solution/root to the problem. But then, why can you divide out things like $tan(x)$ in a problem where we try to prove 2 things are equal, or when trying to prove a trig identity? Isn't that dividing out a potential trig relationship?
algebra-precalculus trigonometry
edited Jul 27 at 0:43
bjcolby15
7921616
asked Jul 26 at 13:44
Ethan Chan
608322
5
For a given $x$, $tan(x)$ is just a number – it could be any number, ahead of time. When is dividing by a number a bad idea?
– gj255
Jul 26 at 13:48
2
@gj255 When that number is 0? Because tan(x)/tan(x)=0/0 undefined?
– Ethan Chan
Jul 26 at 13:49
@gj255 But then why, when proving trig identities and such, can you divide tan(x)/tan(x) in the process of simplifying? There's also the possibility that tan(x)/tan(x)=0/0 ?
– Ethan Chan
Jul 26 at 13:52
4
Exactly. So you should treat the two cases $tan(x) = 0$ and $tan(x) neq 0$ separately, if you want to make sure you've found all solutions. If there is a particular trig identity you're referring to, please include it in your post.
– gj255
Jul 26 at 13:54
1
Rather than thinking about turning equations into other equations, think about "$implies$" and "$iff$". Use them in $every$ step, and do not omit to write them at every step. This will prevent you from becoming muddled in a morass of mysterious methods.
– DanielWainfleet
Jul 26 at 18:21
 |Â
show 4 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The textbook I'm using has this problem involving trig equations which goes $tan(x) sin^2(x)=2 tan(x)$; we're trying to solve for x. The textbook advises not to divide this problem by $tan(x)$ (in other words not to go $tan(x)/tan(x) sin^2(x)=2(tan(x)/tan(x))$.
I don't understand why that won't be a good idea. After all, when we try to prove trig identities (I can't remember which one), don't we also do the same thing, dividing out things like $tan(x)$ and $sin(x)$ in order to simplify it? Or at least in trig problems where we try to prove 2 things are equal? So why can't we do that right here?
I'm guessing it's because by dividing out $tan(x)$ in a problem where you are supposed to find the solution of $x$, you are dividing out a potential solution/root to the problem. But then, why can you divide out things like $tan(x)$ in a problem where we try to prove 2 things are equal, or when trying to prove a trig identity? Isn't that dividing out a potential trig relationship?
algebra-precalculus trigonometry
edited Jul 27 at 0:43
bjcolby15
7921616
asked Jul 26 at 13:44
Ethan Chan
608322
The textbook I'm using has this problem involving trig equations which goes $tan(x) sin^2(x)=2 tan(x)$; we're trying to solve for x. The textbook advises not to divide this problem by $tan(x)$ (in other words not to go $tan(x)/tan(x) sin^2(x)=2(tan(x)/tan(x))$.
I don't understand why that won't be a good idea. After all, when we try to prove trig identities (I can't remember which one), don't we also do the same thing, dividing out things like $tan(x)$ and $sin(x)$ in order to simplify it? Or at least in trig problems where we try to prove 2 things are equal? So why can't we do that right here?
I'm guessing it's because by dividing out $tan(x)$ in a problem where you are supposed to find the solution of $x$, you are dividing out a potential solution/root to the problem. But then, why can you divide out things like $tan(x)$ in a problem where we try to prove 2 things are equal, or when trying to prove a trig identity? Isn't that dividing out a potential trig relationship?
algebra-precalculus trigonometry
edited Jul 27 at 0:43
bjcolby15
7921616
asked Jul 26 at 13:44
Ethan Chan
608322
edited Jul 27 at 0:43
bjcolby15
7921616
edited Jul 27 at 0:43
bjcolby15
7921616
edited Jul 27 at 0:43
bjcolby15
7921616
7921616
asked Jul 26 at 13:44
Ethan Chan
608322
asked Jul 26 at 13:44
Ethan Chan
608322
asked Jul 26 at 13:44
Ethan Chan
608322
608322
5
For a given $x$, $tan(x)$ is just a number – it could be any number, ahead of time. When is dividing by a number a bad idea?
– gj255
Jul 26 at 13:48
2
@gj255 When that number is 0? Because tan(x)/tan(x)=0/0 undefined?
– Ethan Chan
Jul 26 at 13:49
@gj255 But then why, when proving trig identities and such, can you divide tan(x)/tan(x) in the process of simplifying? There's also the possibility that tan(x)/tan(x)=0/0 ?
– Ethan Chan
Jul 26 at 13:52
4
Exactly. So you should treat the two cases $tan(x) = 0$ and $tan(x) neq 0$ separately, if you want to make sure you've found all solutions. If there is a particular trig identity you're referring to, please include it in your post.
– gj255
Jul 26 at 13:54
1
Rather than thinking about turning equations into other equations, think about "$implies$" and "$iff$". Use them in $every$ step, and do not omit to write them at every step. This will prevent you from becoming muddled in a morass of mysterious methods.
– DanielWainfleet
Jul 26 at 18:21
 |Â
show 4 more comments
5
For a given $x$, $tan(x)$ is just a number – it could be any number, ahead of time. When is dividing by a number a bad idea?
– gj255
Jul 26 at 13:48
2
@gj255 When that number is 0? Because tan(x)/tan(x)=0/0 undefined?
– Ethan Chan
Jul 26 at 13:49
@gj255 But then why, when proving trig identities and such, can you divide tan(x)/tan(x) in the process of simplifying? There's also the possibility that tan(x)/tan(x)=0/0 ?
– Ethan Chan
Jul 26 at 13:52
4
Exactly. So you should treat the two cases $tan(x) = 0$ and $tan(x) neq 0$ separately, if you want to make sure you've found all solutions. If there is a particular trig identity you're referring to, please include it in your post.
– gj255
Jul 26 at 13:54
1
Rather than thinking about turning equations into other equations, think about "$implies$" and "$iff$". Use them in $every$ step, and do not omit to write them at every step. This will prevent you from becoming muddled in a morass of mysterious methods.
– DanielWainfleet
Jul 26 at 18:21
5
5
For a given $x$, $tan(x)$ is just a number – it could be any number, ahead of time. When is dividing by a number a bad idea?
– gj255
Jul 26 at 13:48
For a given $x$, $tan(x)$ is just a number – it could be any number, ahead of time. When is dividing by a number a bad idea?
– gj255
Jul 26 at 13:48
2
2
@gj255 When that number is 0? Because tan(x)/tan(x)=0/0 undefined?
– Ethan Chan
Jul 26 at 13:49
@gj255 When that number is 0? Because tan(x)/tan(x)=0/0 undefined?
– Ethan Chan
Jul 26 at 13:49
@gj255 But then why, when proving trig identities and such, can you divide tan(x)/tan(x) in the process of simplifying? There's also the possibility that tan(x)/tan(x)=0/0 ?
– Ethan Chan
Jul 26 at 13:52
@gj255 But then why, when proving trig identities and such, can you divide tan(x)/tan(x) in the process of simplifying? There's also the possibility that tan(x)/tan(x)=0/0 ?
– Ethan Chan
Jul 26 at 13:52
4
4
Exactly. So you should treat the two cases $tan(x) = 0$ and $tan(x) neq 0$ separately, if you want to make sure you've found all solutions. If there is a particular trig identity you're referring to, please include it in your post.
– gj255
Jul 26 at 13:54
Exactly. So you should treat the two cases $tan(x) = 0$ and $tan(x) neq 0$ separately, if you want to make sure you've found all solutions. If there is a particular trig identity you're referring to, please include it in your post.
– gj255
Jul 26 at 13:54
1
1
Rather than thinking about turning equations into other equations, think about "$implies$" and "$iff$". Use them in $every$ step, and do not omit to write them at every step. This will prevent you from becoming muddled in a morass of mysterious methods.
– DanielWainfleet
Jul 26 at 18:21
Rather than thinking about turning equations into other equations, think about "$implies$" and "$iff$". Use them in $every$ step, and do not omit to write them at every step. This will prevent you from becoming muddled in a morass of mysterious methods.
– DanielWainfleet
Jul 26 at 18:21
 |Â
show 4 more comments
2 Answers
2
active
oldest
votes
up vote
3
down vote
accepted
As a rule, one should never divide both sides of an equation by anything, unless it has already been proved that that thing is never $0$. This is the case for $tan x$ in your equation as much as it is for $x$ in the equation $x^2=x$.
The elimination of a $tan x$ has the additional problem of changing the "domain of existence" of the equation. If anywhere in an equation some $tan x$ appears, then the adept student should assume that the implicit hypothesis $frac xpi-frac12notin Bbb Z$ holds throughout the whole discussion.
The first obstruction must be dealt in the usual way: just like no one ever solves $x^2=x$ by dividing both sides by $x$, but rather he just rewrites it in the equivalent form $$x^2=xiff x^2-x=0iff x(x-1)=0$$ and then uses the annulment of product law, you should carry everything to the left and then group the terms.
The way to go around the second obstruction is by taking note of all the domains of existence of the expressions that appear in your equation before starting to work on it. Once you've made clear that the entire discussion (solutions included) is restricted to those domains, algebraic manipulations (the correct ones) are allowed.
answered Jul 26 at 14:03
Saucy O'Path
2,444217
So that means even when proving an identity (basically one of those trig problems where you have to prove something = something else), you can't divide by say tan(x) and the like?
– Ethan Chan
Jul 26 at 14:08
1
@EthanChan Technically, you shouldn't. Often, though, there is the implicit assumption that you'll get full score as long as you obtain an identity that holds in some interval, and even if you don't discuss the domain of validity of the manipulations you are doing.
– Saucy O'Path
Jul 26 at 14:11
add a comment |Â
up vote
1
down vote
If you try to prove an identity and let's say on both sides you have $tan(x)$, in something like $tan(x) f(x)=tan(x)g(x)$ you can use the division. That is because if $tan(x)=0$ the identity is verified. And you want to prove that it's true for all $x$ values, not just the ones where $tan(x)=0$. If you want to find solutions to something like the above, you have solutions when $f(x)=g(x)$, and in addition you have solutions where $tan(x)=0$.
answered Jul 26 at 13:52
Andrei
7,3702822
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
As a rule, one should never divide both sides of an equation by anything, unless it has already been proved that that thing is never $0$. This is the case for $tan x$ in your equation as much as it is for $x$ in the equation $x^2=x$.
The elimination of a $tan x$ has the additional problem of changing the "domain of existence" of the equation. If anywhere in an equation some $tan x$ appears, then the adept student should assume that the implicit hypothesis $frac xpi-frac12notin Bbb Z$ holds throughout the whole discussion.
The first obstruction must be dealt in the usual way: just like no one ever solves $x^2=x$ by dividing both sides by $x$, but rather he just rewrites it in the equivalent form $$x^2=xiff x^2-x=0iff x(x-1)=0$$ and then uses the annulment of product law, you should carry everything to the left and then group the terms.
The way to go around the second obstruction is by taking note of all the domains of existence of the expressions that appear in your equation before starting to work on it. Once you've made clear that the entire discussion (solutions included) is restricted to those domains, algebraic manipulations (the correct ones) are allowed.
answered Jul 26 at 14:03
Saucy O'Path
2,444217
So that means even when proving an identity (basically one of those trig problems where you have to prove something = something else), you can't divide by say tan(x) and the like?
– Ethan Chan
Jul 26 at 14:08
1
@EthanChan Technically, you shouldn't. Often, though, there is the implicit assumption that you'll get full score as long as you obtain an identity that holds in some interval, and even if you don't discuss the domain of validity of the manipulations you are doing.
– Saucy O'Path
Jul 26 at 14:11
add a comment |Â
up vote
3
down vote
accepted
As a rule, one should never divide both sides of an equation by anything, unless it has already been proved that that thing is never $0$. This is the case for $tan x$ in your equation as much as it is for $x$ in the equation $x^2=x$.
The elimination of a $tan x$ has the additional problem of changing the "domain of existence" of the equation. If anywhere in an equation some $tan x$ appears, then the adept student should assume that the implicit hypothesis $frac xpi-frac12notin Bbb Z$ holds throughout the whole discussion.
The first obstruction must be dealt in the usual way: just like no one ever solves $x^2=x$ by dividing both sides by $x$, but rather he just rewrites it in the equivalent form $$x^2=xiff x^2-x=0iff x(x-1)=0$$ and then uses the annulment of product law, you should carry everything to the left and then group the terms.
The way to go around the second obstruction is by taking note of all the domains of existence of the expressions that appear in your equation before starting to work on it. Once you've made clear that the entire discussion (solutions included) is restricted to those domains, algebraic manipulations (the correct ones) are allowed.
answered Jul 26 at 14:03
Saucy O'Path
2,444217
So that means even when proving an identity (basically one of those trig problems where you have to prove something = something else), you can't divide by say tan(x) and the like?
– Ethan Chan
Jul 26 at 14:08
1
@EthanChan Technically, you shouldn't. Often, though, there is the implicit assumption that you'll get full score as long as you obtain an identity that holds in some interval, and even if you don't discuss the domain of validity of the manipulations you are doing.
– Saucy O'Path
Jul 26 at 14:11
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
As a rule, one should never divide both sides of an equation by anything, unless it has already been proved that that thing is never $0$. This is the case for $tan x$ in your equation as much as it is for $x$ in the equation $x^2=x$.
The elimination of a $tan x$ has the additional problem of changing the "domain of existence" of the equation. If anywhere in an equation some $tan x$ appears, then the adept student should assume that the implicit hypothesis $frac xpi-frac12notin Bbb Z$ holds throughout the whole discussion.
The first obstruction must be dealt in the usual way: just like no one ever solves $x^2=x$ by dividing both sides by $x$, but rather he just rewrites it in the equivalent form $$x^2=xiff x^2-x=0iff x(x-1)=0$$ and then uses the annulment of product law, you should carry everything to the left and then group the terms.
The way to go around the second obstruction is by taking note of all the domains of existence of the expressions that appear in your equation before starting to work on it. Once you've made clear that the entire discussion (solutions included) is restricted to those domains, algebraic manipulations (the correct ones) are allowed.
answered Jul 26 at 14:03
Saucy O'Path
2,444217
As a rule, one should never divide both sides of an equation by anything, unless it has already been proved that that thing is never $0$. This is the case for $tan x$ in your equation as much as it is for $x$ in the equation $x^2=x$.
The elimination of a $tan x$ has the additional problem of changing the "domain of existence" of the equation. If anywhere in an equation some $tan x$ appears, then the adept student should assume that the implicit hypothesis $frac xpi-frac12notin Bbb Z$ holds throughout the whole discussion.
The first obstruction must be dealt in the usual way: just like no one ever solves $x^2=x$ by dividing both sides by $x$, but rather he just rewrites it in the equivalent form $$x^2=xiff x^2-x=0iff x(x-1)=0$$ and then uses the annulment of product law, you should carry everything to the left and then group the terms.
The way to go around the second obstruction is by taking note of all the domains of existence of the expressions that appear in your equation before starting to work on it. Once you've made clear that the entire discussion (solutions included) is restricted to those domains, algebraic manipulations (the correct ones) are allowed.
answered Jul 26 at 14:03
Saucy O'Path
2,444217
edited Jul 27 at 7:01
answered Jul 26 at 14:03
Saucy O'Path
2,444217
answered Jul 26 at 14:03
Saucy O'Path
2,444217
answered Jul 26 at 14:03
Saucy O'Path
2,444217
2,444217
So that means even when proving an identity (basically one of those trig problems where you have to prove something = something else), you can't divide by say tan(x) and the like?
– Ethan Chan
Jul 26 at 14:08
1
@EthanChan Technically, you shouldn't. Often, though, there is the implicit assumption that you'll get full score as long as you obtain an identity that holds in some interval, and even if you don't discuss the domain of validity of the manipulations you are doing.
– Saucy O'Path
Jul 26 at 14:11
add a comment |Â
So that means even when proving an identity (basically one of those trig problems where you have to prove something = something else), you can't divide by say tan(x) and the like?
– Ethan Chan
Jul 26 at 14:08
1
@EthanChan Technically, you shouldn't. Often, though, there is the implicit assumption that you'll get full score as long as you obtain an identity that holds in some interval, and even if you don't discuss the domain of validity of the manipulations you are doing.
– Saucy O'Path
Jul 26 at 14:11
So that means even when proving an identity (basically one of those trig problems where you have to prove something = something else), you can't divide by say tan(x) and the like?
– Ethan Chan
Jul 26 at 14:08
So that means even when proving an identity (basically one of those trig problems where you have to prove something = something else), you can't divide by say tan(x) and the like?
– Ethan Chan
Jul 26 at 14:08
1
1
@EthanChan Technically, you shouldn't. Often, though, there is the implicit assumption that you'll get full score as long as you obtain an identity that holds in some interval, and even if you don't discuss the domain of validity of the manipulations you are doing.
– Saucy O'Path
Jul 26 at 14:11
@EthanChan Technically, you shouldn't. Often, though, there is the implicit assumption that you'll get full score as long as you obtain an identity that holds in some interval, and even if you don't discuss the domain of validity of the manipulations you are doing.
– Saucy O'Path
Jul 26 at 14:11
add a comment |Â
up vote
1
down vote
If you try to prove an identity and let's say on both sides you have $tan(x)$, in something like $tan(x) f(x)=tan(x)g(x)$ you can use the division. That is because if $tan(x)=0$ the identity is verified. And you want to prove that it's true for all $x$ values, not just the ones where $tan(x)=0$. If you want to find solutions to something like the above, you have solutions when $f(x)=g(x)$, and in addition you have solutions where $tan(x)=0$.
answered Jul 26 at 13:52
Andrei
7,3702822
add a comment |Â
up vote
1
down vote
If you try to prove an identity and let's say on both sides you have $tan(x)$, in something like $tan(x) f(x)=tan(x)g(x)$ you can use the division. That is because if $tan(x)=0$ the identity is verified. And you want to prove that it's true for all $x$ values, not just the ones where $tan(x)=0$. If you want to find solutions to something like the above, you have solutions when $f(x)=g(x)$, and in addition you have solutions where $tan(x)=0$.
answered Jul 26 at 13:52
Andrei
7,3702822
add a comment |Â
up vote
1
down vote
up vote
1
down vote
If you try to prove an identity and let's say on both sides you have $tan(x)$, in something like $tan(x) f(x)=tan(x)g(x)$ you can use the division. That is because if $tan(x)=0$ the identity is verified. And you want to prove that it's true for all $x$ values, not just the ones where $tan(x)=0$. If you want to find solutions to something like the above, you have solutions when $f(x)=g(x)$, and in addition you have solutions where $tan(x)=0$.
answered Jul 26 at 13:52
Andrei
7,3702822
If you try to prove an identity and let's say on both sides you have $tan(x)$, in something like $tan(x) f(x)=tan(x)g(x)$ you can use the division. That is because if $tan(x)=0$ the identity is verified. And you want to prove that it's true for all $x$ values, not just the ones where $tan(x)=0$. If you want to find solutions to something like the above, you have solutions when $f(x)=g(x)$, and in addition you have solutions where $tan(x)=0$.
answered Jul 26 at 13:52
Andrei
7,3702822
answered Jul 26 at 13:52
Andrei
7,3702822
answered Jul 26 at 13:52
Andrei
7,3702822
answered Jul 26 at 13:52
Andrei
7,3702822
7,3702822
add a comment |Â
add a comment |Â
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5
For a given $x$, $tan(x)$ is just a number – it could be any number, ahead of time. When is dividing by a number a bad idea?
– gj255
Jul 26 at 13:48
2
@gj255 When that number is 0? Because tan(x)/tan(x)=0/0 undefined?
– Ethan Chan
Jul 26 at 13:49
@gj255 But then why, when proving trig identities and such, can you divide tan(x)/tan(x) in the process of simplifying? There's also the possibility that tan(x)/tan(x)=0/0 ?
– Ethan Chan
Jul 26 at 13:52
4
Exactly. So you should treat the two cases $tan(x) = 0$ and $tan(x) neq 0$ separately, if you want to make sure you've found all solutions. If there is a particular trig identity you're referring to, please include it in your post.
– gj255
Jul 26 at 13:54
1
Rather than thinking about turning equations into other equations, think about "$implies$" and "$iff$". Use them in $every$ step, and do not omit to write them at every step. This will prevent you from becoming muddled in a morass of mysterious methods.
– DanielWainfleet
Jul 26 at 18:21