If $tan(A-B)=1$ and $sec(A+B)=2/sqrt{3}$, find the minimum positive value of $B$
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If $tan(A-B)=1$ and $sec(A+B)=2/sqrt{3}$, find the minimum positive value of $B$
I am using $tan(A-B)=1$, so $A-B=npi+pi/4$ and $A+B = 2npipmpi/6$. Solving these I am getting $B =7pi/24$ and $A =37pi/24$.
The book I am refering to has marked the answer as $19pi/24$.
Am I doing something wrong?
trigonometry
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up vote
2
down vote
favorite
If $tan(A-B)=1$ and $sec(A+B)=2/sqrt{3}$, find the minimum positive value of $B$
I am using $tan(A-B)=1$, so $A-B=npi+pi/4$ and $A+B = 2npipmpi/6$. Solving these I am getting $B =7pi/24$ and $A =37pi/24$.
The book I am refering to has marked the answer as $19pi/24$.
Am I doing something wrong?
trigonometry
math.meta.stackexchange.com/questions/5020/…
– KM101
Nov 2 at 14:36
$sec(theta)$ is an even function, so $sec(theta) = sec(-theta)$.
– KM101
Nov 2 at 14:53
Well it looks like your answer works, and your value of $B$ is smaller than theirs, so I'd say you're right.
– Sam Streeter
Nov 2 at 14:54
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
If $tan(A-B)=1$ and $sec(A+B)=2/sqrt{3}$, find the minimum positive value of $B$
I am using $tan(A-B)=1$, so $A-B=npi+pi/4$ and $A+B = 2npipmpi/6$. Solving these I am getting $B =7pi/24$ and $A =37pi/24$.
The book I am refering to has marked the answer as $19pi/24$.
Am I doing something wrong?
trigonometry
If $tan(A-B)=1$ and $sec(A+B)=2/sqrt{3}$, find the minimum positive value of $B$
I am using $tan(A-B)=1$, so $A-B=npi+pi/4$ and $A+B = 2npipmpi/6$. Solving these I am getting $B =7pi/24$ and $A =37pi/24$.
The book I am refering to has marked the answer as $19pi/24$.
Am I doing something wrong?
trigonometry
trigonometry
edited Nov 2 at 15:15
user10354138
6,492623
6,492623
asked Nov 2 at 14:32
user62956
112
112
math.meta.stackexchange.com/questions/5020/…
– KM101
Nov 2 at 14:36
$sec(theta)$ is an even function, so $sec(theta) = sec(-theta)$.
– KM101
Nov 2 at 14:53
Well it looks like your answer works, and your value of $B$ is smaller than theirs, so I'd say you're right.
– Sam Streeter
Nov 2 at 14:54
add a comment |
math.meta.stackexchange.com/questions/5020/…
– KM101
Nov 2 at 14:36
$sec(theta)$ is an even function, so $sec(theta) = sec(-theta)$.
– KM101
Nov 2 at 14:53
Well it looks like your answer works, and your value of $B$ is smaller than theirs, so I'd say you're right.
– Sam Streeter
Nov 2 at 14:54
math.meta.stackexchange.com/questions/5020/…
– KM101
Nov 2 at 14:36
math.meta.stackexchange.com/questions/5020/…
– KM101
Nov 2 at 14:36
$sec(theta)$ is an even function, so $sec(theta) = sec(-theta)$.
– KM101
Nov 2 at 14:53
$sec(theta)$ is an even function, so $sec(theta) = sec(-theta)$.
– KM101
Nov 2 at 14:53
Well it looks like your answer works, and your value of $B$ is smaller than theirs, so I'd say you're right.
– Sam Streeter
Nov 2 at 14:54
Well it looks like your answer works, and your value of $B$ is smaller than theirs, so I'd say you're right.
– Sam Streeter
Nov 2 at 14:54
add a comment |
3 Answers
3
active
oldest
votes
up vote
0
down vote
You correctly have
$$
A=B+frac{pi}{4}+npi
$$
Therefore $A+B=2B+pi/4+npi$. Hence
$$
cosleft(2B+frac{pi}{4}+npiright)=frac{sqrt{3}}{2}=cosfrac{pi}{6}
$$
Therefore either
$$
2B+frac{pi}{4}+npi=frac{pi}{6}+2mpi
$$
or
$$
2B+frac{pi}{4}+npi=-frac{pi}{6}+2mpi
$$
In the first case
$$
2B=-frac{pi}{12}+(2m-n)pi
$$
which is minimal positive for $2m-n=1$, giving
$$
B=frac{11}{24}pi
$$
In the second case,
$$
2B=-frac{5}{12}pi+(2m-n)pi
$$
that gives the minimal positive solution
$$
B=frac{7}{24}pi
$$
You are right and the book is wrong.
add a comment |
up vote
0
down vote
First of all, your values for $A$ and $B$ satisfy the given formulae and $B$ is positive, so their proposed minimum for $B$ is definitely wrong.
One thing you did miss in your reasoning is that there are two families of solutions to $secleft(thetaright) = frac{2}{sqrt{3}}$; there should be a $pm$ with $frac{pi}{6}$ in your formula for $A+B$. However, it turns out that you get the minimum value for $B$ from the family of solutions you gave, and the minimum positive value of $B$ is indeed $frac{7pi}{24}$.
Explicitly, $$A-B = frac{pi}{4} + npi,$$ $$A+B = pm frac{pi}{6} + 2mpi,$$ so $$B = frac{pi}{2}left(pmfrac{1}{6} - frac{1}{4} + left(2m - nright)right),$$ and then we see that $$B = frac{pi}{2}left(-frac{1}{12} + kright), quad k in mathbb{Z}$$ or $$B = frac{pi}{2}left(frac{-5}{12} + kright), quad k in mathbb{Z},$$ and the smallest positive value of $B$ is then $frac{7pi}{24}$.
EDIT: I see that you have/somebody has now edited your question to include the $pm$.
add a comment |
up vote
0
down vote
Your answer is correct. For all $n in mathbb{Z}$:
$$tan(A-B) = 1 implies A-B = frac{pi}{4}+pi n$$
Recall that secant, like cosine, is an even function. (Which you've apparently referred to as well, after the format editing.)
$$sec(-theta) = sec(theta)$$
$$sec(A+B) = frac{2}{sqrt 3} implies cos(A+B) = frac{sqrt 3}{2} implies pm(A+B) = frac{pi}{6}+2pi n$$
Therefore, there are $2$ cases, neither of which yields the book's answer.
Case $1$: With $+(A+B)$, by subtracting the two equations to eliminate $A$, the following equation is obtained.
$$-2B = frac{pi}{4}+pi n-bigg(frac{pi}{6}+2pi nbigg)$$
$$-2B = frac{pi}{4}+pi n-frac{pi}{6}-2pi n$$
$$-2B = frac{pi}{12}-pi n$$
$$B = frac{pi n}{2}-frac{pi}{12}$$
The minimum here is $B = frac{11pi}{24}$.
Case $2$: With $-(A+B)$, by adding the two equations to eliminate $A$, the following equation is obtained.
$$-2B = frac{pi}{4}+pi n+bigg(frac{pi}{6}+2pi nbigg)$$
$$-2B = frac{pi}{4}+pi n+frac{pi}{6}+2pi n$$
$$-2B = frac{5pi}{12}+3pi n$$
$$B = -frac{3pi n}{2}-frac{5pi}{24}$$
The minimum here is $B = frac{7pi}{24}$.
Thus, the minimum positive value of $B$, as you correctly found, is $frac{7pi}{24}$.
add a comment |
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
You correctly have
$$
A=B+frac{pi}{4}+npi
$$
Therefore $A+B=2B+pi/4+npi$. Hence
$$
cosleft(2B+frac{pi}{4}+npiright)=frac{sqrt{3}}{2}=cosfrac{pi}{6}
$$
Therefore either
$$
2B+frac{pi}{4}+npi=frac{pi}{6}+2mpi
$$
or
$$
2B+frac{pi}{4}+npi=-frac{pi}{6}+2mpi
$$
In the first case
$$
2B=-frac{pi}{12}+(2m-n)pi
$$
which is minimal positive for $2m-n=1$, giving
$$
B=frac{11}{24}pi
$$
In the second case,
$$
2B=-frac{5}{12}pi+(2m-n)pi
$$
that gives the minimal positive solution
$$
B=frac{7}{24}pi
$$
You are right and the book is wrong.
add a comment |
up vote
0
down vote
You correctly have
$$
A=B+frac{pi}{4}+npi
$$
Therefore $A+B=2B+pi/4+npi$. Hence
$$
cosleft(2B+frac{pi}{4}+npiright)=frac{sqrt{3}}{2}=cosfrac{pi}{6}
$$
Therefore either
$$
2B+frac{pi}{4}+npi=frac{pi}{6}+2mpi
$$
or
$$
2B+frac{pi}{4}+npi=-frac{pi}{6}+2mpi
$$
In the first case
$$
2B=-frac{pi}{12}+(2m-n)pi
$$
which is minimal positive for $2m-n=1$, giving
$$
B=frac{11}{24}pi
$$
In the second case,
$$
2B=-frac{5}{12}pi+(2m-n)pi
$$
that gives the minimal positive solution
$$
B=frac{7}{24}pi
$$
You are right and the book is wrong.
add a comment |
up vote
0
down vote
up vote
0
down vote
You correctly have
$$
A=B+frac{pi}{4}+npi
$$
Therefore $A+B=2B+pi/4+npi$. Hence
$$
cosleft(2B+frac{pi}{4}+npiright)=frac{sqrt{3}}{2}=cosfrac{pi}{6}
$$
Therefore either
$$
2B+frac{pi}{4}+npi=frac{pi}{6}+2mpi
$$
or
$$
2B+frac{pi}{4}+npi=-frac{pi}{6}+2mpi
$$
In the first case
$$
2B=-frac{pi}{12}+(2m-n)pi
$$
which is minimal positive for $2m-n=1$, giving
$$
B=frac{11}{24}pi
$$
In the second case,
$$
2B=-frac{5}{12}pi+(2m-n)pi
$$
that gives the minimal positive solution
$$
B=frac{7}{24}pi
$$
You are right and the book is wrong.
You correctly have
$$
A=B+frac{pi}{4}+npi
$$
Therefore $A+B=2B+pi/4+npi$. Hence
$$
cosleft(2B+frac{pi}{4}+npiright)=frac{sqrt{3}}{2}=cosfrac{pi}{6}
$$
Therefore either
$$
2B+frac{pi}{4}+npi=frac{pi}{6}+2mpi
$$
or
$$
2B+frac{pi}{4}+npi=-frac{pi}{6}+2mpi
$$
In the first case
$$
2B=-frac{pi}{12}+(2m-n)pi
$$
which is minimal positive for $2m-n=1$, giving
$$
B=frac{11}{24}pi
$$
In the second case,
$$
2B=-frac{5}{12}pi+(2m-n)pi
$$
that gives the minimal positive solution
$$
B=frac{7}{24}pi
$$
You are right and the book is wrong.
answered Nov 2 at 15:03
egreg
174k1383198
174k1383198
add a comment |
add a comment |
up vote
0
down vote
First of all, your values for $A$ and $B$ satisfy the given formulae and $B$ is positive, so their proposed minimum for $B$ is definitely wrong.
One thing you did miss in your reasoning is that there are two families of solutions to $secleft(thetaright) = frac{2}{sqrt{3}}$; there should be a $pm$ with $frac{pi}{6}$ in your formula for $A+B$. However, it turns out that you get the minimum value for $B$ from the family of solutions you gave, and the minimum positive value of $B$ is indeed $frac{7pi}{24}$.
Explicitly, $$A-B = frac{pi}{4} + npi,$$ $$A+B = pm frac{pi}{6} + 2mpi,$$ so $$B = frac{pi}{2}left(pmfrac{1}{6} - frac{1}{4} + left(2m - nright)right),$$ and then we see that $$B = frac{pi}{2}left(-frac{1}{12} + kright), quad k in mathbb{Z}$$ or $$B = frac{pi}{2}left(frac{-5}{12} + kright), quad k in mathbb{Z},$$ and the smallest positive value of $B$ is then $frac{7pi}{24}$.
EDIT: I see that you have/somebody has now edited your question to include the $pm$.
add a comment |
up vote
0
down vote
First of all, your values for $A$ and $B$ satisfy the given formulae and $B$ is positive, so their proposed minimum for $B$ is definitely wrong.
One thing you did miss in your reasoning is that there are two families of solutions to $secleft(thetaright) = frac{2}{sqrt{3}}$; there should be a $pm$ with $frac{pi}{6}$ in your formula for $A+B$. However, it turns out that you get the minimum value for $B$ from the family of solutions you gave, and the minimum positive value of $B$ is indeed $frac{7pi}{24}$.
Explicitly, $$A-B = frac{pi}{4} + npi,$$ $$A+B = pm frac{pi}{6} + 2mpi,$$ so $$B = frac{pi}{2}left(pmfrac{1}{6} - frac{1}{4} + left(2m - nright)right),$$ and then we see that $$B = frac{pi}{2}left(-frac{1}{12} + kright), quad k in mathbb{Z}$$ or $$B = frac{pi}{2}left(frac{-5}{12} + kright), quad k in mathbb{Z},$$ and the smallest positive value of $B$ is then $frac{7pi}{24}$.
EDIT: I see that you have/somebody has now edited your question to include the $pm$.
add a comment |
up vote
0
down vote
up vote
0
down vote
First of all, your values for $A$ and $B$ satisfy the given formulae and $B$ is positive, so their proposed minimum for $B$ is definitely wrong.
One thing you did miss in your reasoning is that there are two families of solutions to $secleft(thetaright) = frac{2}{sqrt{3}}$; there should be a $pm$ with $frac{pi}{6}$ in your formula for $A+B$. However, it turns out that you get the minimum value for $B$ from the family of solutions you gave, and the minimum positive value of $B$ is indeed $frac{7pi}{24}$.
Explicitly, $$A-B = frac{pi}{4} + npi,$$ $$A+B = pm frac{pi}{6} + 2mpi,$$ so $$B = frac{pi}{2}left(pmfrac{1}{6} - frac{1}{4} + left(2m - nright)right),$$ and then we see that $$B = frac{pi}{2}left(-frac{1}{12} + kright), quad k in mathbb{Z}$$ or $$B = frac{pi}{2}left(frac{-5}{12} + kright), quad k in mathbb{Z},$$ and the smallest positive value of $B$ is then $frac{7pi}{24}$.
EDIT: I see that you have/somebody has now edited your question to include the $pm$.
First of all, your values for $A$ and $B$ satisfy the given formulae and $B$ is positive, so their proposed minimum for $B$ is definitely wrong.
One thing you did miss in your reasoning is that there are two families of solutions to $secleft(thetaright) = frac{2}{sqrt{3}}$; there should be a $pm$ with $frac{pi}{6}$ in your formula for $A+B$. However, it turns out that you get the minimum value for $B$ from the family of solutions you gave, and the minimum positive value of $B$ is indeed $frac{7pi}{24}$.
Explicitly, $$A-B = frac{pi}{4} + npi,$$ $$A+B = pm frac{pi}{6} + 2mpi,$$ so $$B = frac{pi}{2}left(pmfrac{1}{6} - frac{1}{4} + left(2m - nright)right),$$ and then we see that $$B = frac{pi}{2}left(-frac{1}{12} + kright), quad k in mathbb{Z}$$ or $$B = frac{pi}{2}left(frac{-5}{12} + kright), quad k in mathbb{Z},$$ and the smallest positive value of $B$ is then $frac{7pi}{24}$.
EDIT: I see that you have/somebody has now edited your question to include the $pm$.
edited Nov 2 at 15:27
answered Nov 2 at 15:01
Sam Streeter
1,481417
1,481417
add a comment |
add a comment |
up vote
0
down vote
Your answer is correct. For all $n in mathbb{Z}$:
$$tan(A-B) = 1 implies A-B = frac{pi}{4}+pi n$$
Recall that secant, like cosine, is an even function. (Which you've apparently referred to as well, after the format editing.)
$$sec(-theta) = sec(theta)$$
$$sec(A+B) = frac{2}{sqrt 3} implies cos(A+B) = frac{sqrt 3}{2} implies pm(A+B) = frac{pi}{6}+2pi n$$
Therefore, there are $2$ cases, neither of which yields the book's answer.
Case $1$: With $+(A+B)$, by subtracting the two equations to eliminate $A$, the following equation is obtained.
$$-2B = frac{pi}{4}+pi n-bigg(frac{pi}{6}+2pi nbigg)$$
$$-2B = frac{pi}{4}+pi n-frac{pi}{6}-2pi n$$
$$-2B = frac{pi}{12}-pi n$$
$$B = frac{pi n}{2}-frac{pi}{12}$$
The minimum here is $B = frac{11pi}{24}$.
Case $2$: With $-(A+B)$, by adding the two equations to eliminate $A$, the following equation is obtained.
$$-2B = frac{pi}{4}+pi n+bigg(frac{pi}{6}+2pi nbigg)$$
$$-2B = frac{pi}{4}+pi n+frac{pi}{6}+2pi n$$
$$-2B = frac{5pi}{12}+3pi n$$
$$B = -frac{3pi n}{2}-frac{5pi}{24}$$
The minimum here is $B = frac{7pi}{24}$.
Thus, the minimum positive value of $B$, as you correctly found, is $frac{7pi}{24}$.
add a comment |
up vote
0
down vote
Your answer is correct. For all $n in mathbb{Z}$:
$$tan(A-B) = 1 implies A-B = frac{pi}{4}+pi n$$
Recall that secant, like cosine, is an even function. (Which you've apparently referred to as well, after the format editing.)
$$sec(-theta) = sec(theta)$$
$$sec(A+B) = frac{2}{sqrt 3} implies cos(A+B) = frac{sqrt 3}{2} implies pm(A+B) = frac{pi}{6}+2pi n$$
Therefore, there are $2$ cases, neither of which yields the book's answer.
Case $1$: With $+(A+B)$, by subtracting the two equations to eliminate $A$, the following equation is obtained.
$$-2B = frac{pi}{4}+pi n-bigg(frac{pi}{6}+2pi nbigg)$$
$$-2B = frac{pi}{4}+pi n-frac{pi}{6}-2pi n$$
$$-2B = frac{pi}{12}-pi n$$
$$B = frac{pi n}{2}-frac{pi}{12}$$
The minimum here is $B = frac{11pi}{24}$.
Case $2$: With $-(A+B)$, by adding the two equations to eliminate $A$, the following equation is obtained.
$$-2B = frac{pi}{4}+pi n+bigg(frac{pi}{6}+2pi nbigg)$$
$$-2B = frac{pi}{4}+pi n+frac{pi}{6}+2pi n$$
$$-2B = frac{5pi}{12}+3pi n$$
$$B = -frac{3pi n}{2}-frac{5pi}{24}$$
The minimum here is $B = frac{7pi}{24}$.
Thus, the minimum positive value of $B$, as you correctly found, is $frac{7pi}{24}$.
add a comment |
up vote
0
down vote
up vote
0
down vote
Your answer is correct. For all $n in mathbb{Z}$:
$$tan(A-B) = 1 implies A-B = frac{pi}{4}+pi n$$
Recall that secant, like cosine, is an even function. (Which you've apparently referred to as well, after the format editing.)
$$sec(-theta) = sec(theta)$$
$$sec(A+B) = frac{2}{sqrt 3} implies cos(A+B) = frac{sqrt 3}{2} implies pm(A+B) = frac{pi}{6}+2pi n$$
Therefore, there are $2$ cases, neither of which yields the book's answer.
Case $1$: With $+(A+B)$, by subtracting the two equations to eliminate $A$, the following equation is obtained.
$$-2B = frac{pi}{4}+pi n-bigg(frac{pi}{6}+2pi nbigg)$$
$$-2B = frac{pi}{4}+pi n-frac{pi}{6}-2pi n$$
$$-2B = frac{pi}{12}-pi n$$
$$B = frac{pi n}{2}-frac{pi}{12}$$
The minimum here is $B = frac{11pi}{24}$.
Case $2$: With $-(A+B)$, by adding the two equations to eliminate $A$, the following equation is obtained.
$$-2B = frac{pi}{4}+pi n+bigg(frac{pi}{6}+2pi nbigg)$$
$$-2B = frac{pi}{4}+pi n+frac{pi}{6}+2pi n$$
$$-2B = frac{5pi}{12}+3pi n$$
$$B = -frac{3pi n}{2}-frac{5pi}{24}$$
The minimum here is $B = frac{7pi}{24}$.
Thus, the minimum positive value of $B$, as you correctly found, is $frac{7pi}{24}$.
Your answer is correct. For all $n in mathbb{Z}$:
$$tan(A-B) = 1 implies A-B = frac{pi}{4}+pi n$$
Recall that secant, like cosine, is an even function. (Which you've apparently referred to as well, after the format editing.)
$$sec(-theta) = sec(theta)$$
$$sec(A+B) = frac{2}{sqrt 3} implies cos(A+B) = frac{sqrt 3}{2} implies pm(A+B) = frac{pi}{6}+2pi n$$
Therefore, there are $2$ cases, neither of which yields the book's answer.
Case $1$: With $+(A+B)$, by subtracting the two equations to eliminate $A$, the following equation is obtained.
$$-2B = frac{pi}{4}+pi n-bigg(frac{pi}{6}+2pi nbigg)$$
$$-2B = frac{pi}{4}+pi n-frac{pi}{6}-2pi n$$
$$-2B = frac{pi}{12}-pi n$$
$$B = frac{pi n}{2}-frac{pi}{12}$$
The minimum here is $B = frac{11pi}{24}$.
Case $2$: With $-(A+B)$, by adding the two equations to eliminate $A$, the following equation is obtained.
$$-2B = frac{pi}{4}+pi n+bigg(frac{pi}{6}+2pi nbigg)$$
$$-2B = frac{pi}{4}+pi n+frac{pi}{6}+2pi n$$
$$-2B = frac{5pi}{12}+3pi n$$
$$B = -frac{3pi n}{2}-frac{5pi}{24}$$
The minimum here is $B = frac{7pi}{24}$.
Thus, the minimum positive value of $B$, as you correctly found, is $frac{7pi}{24}$.
edited Nov 23 at 9:14
answered Nov 2 at 15:19
KM101
2,527416
2,527416
add a comment |
add a comment |
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math.meta.stackexchange.com/questions/5020/…
– KM101
Nov 2 at 14:36
$sec(theta)$ is an even function, so $sec(theta) = sec(-theta)$.
– KM101
Nov 2 at 14:53
Well it looks like your answer works, and your value of $B$ is smaller than theirs, so I'd say you're right.
– Sam Streeter
Nov 2 at 14:54