Solve $tan (x-fracpi4)=-tan(x+fracpi2)$

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up vote
1
down vote

favorite

1

The question:

Without the use of a calculator, solve for all values of $x$ if $tan (x-fracpi4)=-tan(x+fracpi2).$

Using the compound angle formula for solving equations is normally easy, but I stumbled across this problem.

The $LHS$ is easy to expand, but when you apply the compound formula for the $RHS$,

beginalign
tan(x+fracpi2) & = fractan(x) + tan(fracpi2)1-tan(x)cdottan(fracpi2) \
endalign
You might notice that this is a problem because I cannot evaluate $tan(fracpi2)$. So this is what I tried. First I tried writing

beginalign
tan(x+fracpi2) & = fracsin(x+fracpi2)cos(x+fracpi2) \
& = fraccos (x)sin (x)
endalign
which I knew was wrong. Anyone know how to get around this?

algebra-precalculus trigonometry

share|cite|improve this question

edited Jul 19 at 13:10

asked Jul 19 at 13:04

Landuros

1,7301520

• 
  
  
  

  
  

  
  
  
  
  
  

  
  why don't you make a substitution $y=x-pi/4$?
  – Vasya
  Jul 19 at 13:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Use $-tanx=tan-x$ because $tan$ is an odd function.
  – Shrey Joshi
  Jul 19 at 13:27
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Stop using compound angle formulas of any kind, this problem is completely elementary! Don't use heavy artilery to kill an ant.
  – Oldboy
  Jul 19 at 13:28
  
  

  
  
  
  

add a comment | 

up vote
1
down vote

favorite

1

The question:

Without the use of a calculator, solve for all values of $x$ if $tan (x-fracpi4)=-tan(x+fracpi2).$

Using the compound angle formula for solving equations is normally easy, but I stumbled across this problem.

The $LHS$ is easy to expand, but when you apply the compound formula for the $RHS$,

beginalign
tan(x+fracpi2) & = fractan(x) + tan(fracpi2)1-tan(x)cdottan(fracpi2) \
endalign
You might notice that this is a problem because I cannot evaluate $tan(fracpi2)$. So this is what I tried. First I tried writing

beginalign
tan(x+fracpi2) & = fracsin(x+fracpi2)cos(x+fracpi2) \
& = fraccos (x)sin (x)
endalign
which I knew was wrong. Anyone know how to get around this?

algebra-precalculus trigonometry

share|cite|improve this question

edited Jul 19 at 13:10

asked Jul 19 at 13:04

Landuros

1,7301520

• 
  
  
  

  
  

  
  
  
  
  
  

  
  why don't you make a substitution $y=x-pi/4$?
  – Vasya
  Jul 19 at 13:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Use $-tanx=tan-x$ because $tan$ is an odd function.
  – Shrey Joshi
  Jul 19 at 13:27
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Stop using compound angle formulas of any kind, this problem is completely elementary! Don't use heavy artilery to kill an ant.
  – Oldboy
  Jul 19 at 13:28
  
  

  
  
  
  

add a comment | 

up vote
1
down vote

favorite

1

up vote
1
down vote

favorite

1

1

The question:

Without the use of a calculator, solve for all values of $x$ if $tan (x-fracpi4)=-tan(x+fracpi2).$

Using the compound angle formula for solving equations is normally easy, but I stumbled across this problem.

The $LHS$ is easy to expand, but when you apply the compound formula for the $RHS$,

beginalign
tan(x+fracpi2) & = fractan(x) + tan(fracpi2)1-tan(x)cdottan(fracpi2) \
endalign
You might notice that this is a problem because I cannot evaluate $tan(fracpi2)$. So this is what I tried. First I tried writing

beginalign
tan(x+fracpi2) & = fracsin(x+fracpi2)cos(x+fracpi2) \
& = fraccos (x)sin (x)
endalign
which I knew was wrong. Anyone know how to get around this?

algebra-precalculus trigonometry

share|cite|improve this question

edited Jul 19 at 13:10

asked Jul 19 at 13:04

Landuros

1,7301520

The question:

Without the use of a calculator, solve for all values of $x$ if $tan (x-fracpi4)=-tan(x+fracpi2).$

Using the compound angle formula for solving equations is normally easy, but I stumbled across this problem.

The $LHS$ is easy to expand, but when you apply the compound formula for the $RHS$,

beginalign
tan(x+fracpi2) & = fractan(x) + tan(fracpi2)1-tan(x)cdottan(fracpi2) \
endalign
You might notice that this is a problem because I cannot evaluate $tan(fracpi2)$. So this is what I tried. First I tried writing

beginalign
tan(x+fracpi2) & = fracsin(x+fracpi2)cos(x+fracpi2) \
& = fraccos (x)sin (x)
endalign
which I knew was wrong. Anyone know how to get around this?

algebra-precalculus trigonometry

share|cite|improve this question

edited Jul 19 at 13:10

asked Jul 19 at 13:04

Landuros

1,7301520

share|cite|improve this question

share|cite|improve this question

edited Jul 19 at 13:10

edited Jul 19 at 13:10

edited Jul 19 at 13:10

asked Jul 19 at 13:04

Landuros

1,7301520

asked Jul 19 at 13:04

Landuros

1,7301520

asked Jul 19 at 13:04

Landuros

1,7301520

1,7301520

• 
  
  
  

  
  

  
  
  
  
  
  

  
  why don't you make a substitution $y=x-pi/4$?
  – Vasya
  Jul 19 at 13:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Use $-tanx=tan-x$ because $tan$ is an odd function.
  – Shrey Joshi
  Jul 19 at 13:27
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Stop using compound angle formulas of any kind, this problem is completely elementary! Don't use heavy artilery to kill an ant.
  – Oldboy
  Jul 19 at 13:28
  
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  why don't you make a substitution $y=x-pi/4$?
  – Vasya
  Jul 19 at 13:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Use $-tanx=tan-x$ because $tan$ is an odd function.
  – Shrey Joshi
  Jul 19 at 13:27
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Stop using compound angle formulas of any kind, this problem is completely elementary! Don't use heavy artilery to kill an ant.
  – Oldboy
  Jul 19 at 13:28
  
  

  
  
  
  

why don't you make a substitution $y=x-pi/4$?
– Vasya
Jul 19 at 13:17

why don't you make a substitution $y=x-pi/4$?
– Vasya
Jul 19 at 13:17

Use $-tanx=tan-x$ because $tan$ is an odd function.
– Shrey Joshi
Jul 19 at 13:27

Use $-tanx=tan-x$ because $tan$ is an odd function.
– Shrey Joshi
Jul 19 at 13:27

Stop using compound angle formulas of any kind, this problem is completely elementary! Don't use heavy artilery to kill an ant.
– Oldboy
Jul 19 at 13:28

Stop using compound angle formulas of any kind, this problem is completely elementary! Don't use heavy artilery to kill an ant.
– Oldboy
Jul 19 at 13:28

add a comment | 

4 Answers
4

active

oldest

votes

up vote
3
down vote



accepted

$$tan left(x - fracpi4right) = - tan left(x + fracpi2right)$$

$$tan left(x - fracpi4right) = tan left(-x - fracpi2right)$$

$$x-fracpi4=-x-fracpi2+kpi,quad(kin Z)$$

$$2x=-fracpi4+kpi$$

$$x=-fracpi8+frackpi2$$

Valid for any $kin Z$.

share|cite|improve this answer

edited Jul 19 at 16:17

Robert Soupe

10k21947

answered Jul 19 at 13:22

Oldboy

2,6091316

add a comment | 

up vote
1
down vote

Hint: Use that

$$tan(x)+tan(y)=sec(x)sec(y)sin(x+y)$$

share|cite|improve this answer

answered Jul 19 at 13:23

Dr. Sonnhard Graubner

66.8k32659

• 
  
  
  

  
  

  
  
  
  
  
  

  
  This is probably the most complicated way to solve this equation.
  – Oldboy
  Jul 19 at 13:26
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Nice, you have used the word "probably", probably i will win in LOTO tomorrow.
  – Dr. Sonnhard Graubner
  Jul 19 at 13:36
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Not probably but certainly if you pick the right numbers: 1, 2, 3, 5, 8, 13, 21 and 34.
  – Oldboy
  Jul 19 at 13:55
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Ok, if i will win the price, i would like to share with you
  – Dr. Sonnhard Graubner
  Jul 19 at 14:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I would like it too :)
  – Oldboy
  Jul 19 at 14:07
  

  
  
  
  

 | 
show 2 more comments

up vote
1
down vote

Hint: $tan(x)$ is an odd function so $tan(-x)=-tan(x)$

Solution:

By above we get $tan(x-pi/4)=tan(-x-pi/2)$ so
$$x-pi/4 equiv -x-pi/2 pmodpi$$
(since the tangent function has a period of $pi$)
$$2x equiv -fracpi4 equiv frac3pi4 equiv frac7pi4pmodpi$$
Solving this yields solutions $x=pi n +frac7pi8$ and $x=pi n +frac3pi8$

share|cite|improve this answer

edited Jul 19 at 13:25

answered Jul 19 at 13:18

Shrey Joshi

1389

add a comment | 

up vote
0
down vote

Hint

$$tan x+tan y=fracsin(x+y)cos xcos y$$

share|cite|improve this answer

answered Jul 19 at 13:26

Arnaldo

18k42146

add a comment | 

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4 Answers
4

active

oldest

votes

4 Answers
4

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
3
down vote



accepted

$$tan left(x - fracpi4right) = - tan left(x + fracpi2right)$$

$$tan left(x - fracpi4right) = tan left(-x - fracpi2right)$$

$$x-fracpi4=-x-fracpi2+kpi,quad(kin Z)$$

$$2x=-fracpi4+kpi$$

$$x=-fracpi8+frackpi2$$

Valid for any $kin Z$.

share|cite|improve this answer

edited Jul 19 at 16:17

Robert Soupe

10k21947

answered Jul 19 at 13:22

Oldboy

2,6091316

add a comment | 

up vote
3
down vote



accepted

$$tan left(x - fracpi4right) = - tan left(x + fracpi2right)$$

$$tan left(x - fracpi4right) = tan left(-x - fracpi2right)$$

$$x-fracpi4=-x-fracpi2+kpi,quad(kin Z)$$

$$2x=-fracpi4+kpi$$

$$x=-fracpi8+frackpi2$$

Valid for any $kin Z$.

share|cite|improve this answer

edited Jul 19 at 16:17

Robert Soupe

10k21947

answered Jul 19 at 13:22

Oldboy

2,6091316

add a comment | 

up vote
3
down vote



accepted

up vote
3
down vote



accepted

$$tan left(x - fracpi4right) = - tan left(x + fracpi2right)$$

$$tan left(x - fracpi4right) = tan left(-x - fracpi2right)$$

$$x-fracpi4=-x-fracpi2+kpi,quad(kin Z)$$

$$2x=-fracpi4+kpi$$

$$x=-fracpi8+frackpi2$$

Valid for any $kin Z$.

share|cite|improve this answer

edited Jul 19 at 16:17

Robert Soupe

10k21947

answered Jul 19 at 13:22

Oldboy

2,6091316

$$tan left(x - fracpi4right) = - tan left(x + fracpi2right)$$

$$tan left(x - fracpi4right) = tan left(-x - fracpi2right)$$

$$x-fracpi4=-x-fracpi2+kpi,quad(kin Z)$$

$$2x=-fracpi4+kpi$$

$$x=-fracpi8+frackpi2$$

Valid for any $kin Z$.

share|cite|improve this answer

edited Jul 19 at 16:17

Robert Soupe

10k21947

answered Jul 19 at 13:22

Oldboy

2,6091316

share|cite|improve this answer

share|cite|improve this answer

edited Jul 19 at 16:17

Robert Soupe

10k21947

edited Jul 19 at 16:17

Robert Soupe

10k21947

edited Jul 19 at 16:17

Robert Soupe

10k21947

10k21947

answered Jul 19 at 13:22

Oldboy

2,6091316

answered Jul 19 at 13:22

Oldboy

2,6091316

answered Jul 19 at 13:22

Oldboy

2,6091316

2,6091316

add a comment | 

add a comment | 

up vote
1
down vote

Hint: Use that

$$tan(x)+tan(y)=sec(x)sec(y)sin(x+y)$$

share|cite|improve this answer

answered Jul 19 at 13:23

Dr. Sonnhard Graubner

66.8k32659

• 
  
  
  

  
  

  
  
  
  
  
  

  
  This is probably the most complicated way to solve this equation.
  – Oldboy
  Jul 19 at 13:26
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Nice, you have used the word "probably", probably i will win in LOTO tomorrow.
  – Dr. Sonnhard Graubner
  Jul 19 at 13:36
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Not probably but certainly if you pick the right numbers: 1, 2, 3, 5, 8, 13, 21 and 34.
  – Oldboy
  Jul 19 at 13:55
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Ok, if i will win the price, i would like to share with you
  – Dr. Sonnhard Graubner
  Jul 19 at 14:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I would like it too :)
  – Oldboy
  Jul 19 at 14:07
  

  
  
  
  

 | 
show 2 more comments

up vote
1
down vote

Hint: Use that

$$tan(x)+tan(y)=sec(x)sec(y)sin(x+y)$$

share|cite|improve this answer

answered Jul 19 at 13:23

Dr. Sonnhard Graubner

66.8k32659

• 
  
  
  

  
  

  
  
  
  
  
  

  
  This is probably the most complicated way to solve this equation.
  – Oldboy
  Jul 19 at 13:26
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Nice, you have used the word "probably", probably i will win in LOTO tomorrow.
  – Dr. Sonnhard Graubner
  Jul 19 at 13:36
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Not probably but certainly if you pick the right numbers: 1, 2, 3, 5, 8, 13, 21 and 34.
  – Oldboy
  Jul 19 at 13:55
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Ok, if i will win the price, i would like to share with you
  – Dr. Sonnhard Graubner
  Jul 19 at 14:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I would like it too :)
  – Oldboy
  Jul 19 at 14:07
  

  
  
  
  

 | 
show 2 more comments

up vote
1
down vote

up vote
1
down vote

Hint: Use that

$$tan(x)+tan(y)=sec(x)sec(y)sin(x+y)$$

share|cite|improve this answer

answered Jul 19 at 13:23

Dr. Sonnhard Graubner

66.8k32659

Hint: Use that

$$tan(x)+tan(y)=sec(x)sec(y)sin(x+y)$$

share|cite|improve this answer

answered Jul 19 at 13:23

Dr. Sonnhard Graubner

66.8k32659

share|cite|improve this answer

share|cite|improve this answer

answered Jul 19 at 13:23

Dr. Sonnhard Graubner

66.8k32659

answered Jul 19 at 13:23

Dr. Sonnhard Graubner

66.8k32659

answered Jul 19 at 13:23

Dr. Sonnhard Graubner

66.8k32659

66.8k32659

• 
  
  
  

  
  

  
  
  
  
  
  

  
  This is probably the most complicated way to solve this equation.
  – Oldboy
  Jul 19 at 13:26
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Nice, you have used the word "probably", probably i will win in LOTO tomorrow.
  – Dr. Sonnhard Graubner
  Jul 19 at 13:36
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Not probably but certainly if you pick the right numbers: 1, 2, 3, 5, 8, 13, 21 and 34.
  – Oldboy
  Jul 19 at 13:55
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Ok, if i will win the price, i would like to share with you
  – Dr. Sonnhard Graubner
  Jul 19 at 14:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I would like it too :)
  – Oldboy
  Jul 19 at 14:07
  

  
  
  
  

 | 
show 2 more comments

• 
  
  
  

  
  

  
  
  
  
  
  

  
  This is probably the most complicated way to solve this equation.
  – Oldboy
  Jul 19 at 13:26
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Nice, you have used the word "probably", probably i will win in LOTO tomorrow.
  – Dr. Sonnhard Graubner
  Jul 19 at 13:36
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Not probably but certainly if you pick the right numbers: 1, 2, 3, 5, 8, 13, 21 and 34.
  – Oldboy
  Jul 19 at 13:55
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Ok, if i will win the price, i would like to share with you
  – Dr. Sonnhard Graubner
  Jul 19 at 14:04
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I would like it too :)
  – Oldboy
  Jul 19 at 14:07
  

  
  
  
  

This is probably the most complicated way to solve this equation.
– Oldboy
Jul 19 at 13:26

This is probably the most complicated way to solve this equation.
– Oldboy
Jul 19 at 13:26

1

1

Nice, you have used the word "probably", probably i will win in LOTO tomorrow.
– Dr. Sonnhard Graubner
Jul 19 at 13:36

Nice, you have used the word "probably", probably i will win in LOTO tomorrow.
– Dr. Sonnhard Graubner
Jul 19 at 13:36

Not probably but certainly if you pick the right numbers: 1, 2, 3, 5, 8, 13, 21 and 34.
– Oldboy
Jul 19 at 13:55

Not probably but certainly if you pick the right numbers: 1, 2, 3, 5, 8, 13, 21 and 34.
– Oldboy
Jul 19 at 13:55

1

1

Ok, if i will win the price, i would like to share with you
– Dr. Sonnhard Graubner
Jul 19 at 14:04

Ok, if i will win the price, i would like to share with you
– Dr. Sonnhard Graubner
Jul 19 at 14:04

I would like it too :)
– Oldboy
Jul 19 at 14:07

I would like it too :)
– Oldboy
Jul 19 at 14:07

 | 
show 2 more comments

up vote
1
down vote

Hint: $tan(x)$ is an odd function so $tan(-x)=-tan(x)$

Solution:

By above we get $tan(x-pi/4)=tan(-x-pi/2)$ so
$$x-pi/4 equiv -x-pi/2 pmodpi$$
(since the tangent function has a period of $pi$)
$$2x equiv -fracpi4 equiv frac3pi4 equiv frac7pi4pmodpi$$
Solving this yields solutions $x=pi n +frac7pi8$ and $x=pi n +frac3pi8$

share|cite|improve this answer

edited Jul 19 at 13:25

answered Jul 19 at 13:18

Shrey Joshi

1389

add a comment | 

up vote
1
down vote

Hint: $tan(x)$ is an odd function so $tan(-x)=-tan(x)$

Solution:

By above we get $tan(x-pi/4)=tan(-x-pi/2)$ so
$$x-pi/4 equiv -x-pi/2 pmodpi$$
(since the tangent function has a period of $pi$)
$$2x equiv -fracpi4 equiv frac3pi4 equiv frac7pi4pmodpi$$
Solving this yields solutions $x=pi n +frac7pi8$ and $x=pi n +frac3pi8$

share|cite|improve this answer

edited Jul 19 at 13:25

answered Jul 19 at 13:18

Shrey Joshi

1389

add a comment | 

up vote
1
down vote

up vote
1
down vote

Hint: $tan(x)$ is an odd function so $tan(-x)=-tan(x)$

Solution:

By above we get $tan(x-pi/4)=tan(-x-pi/2)$ so
$$x-pi/4 equiv -x-pi/2 pmodpi$$
(since the tangent function has a period of $pi$)
$$2x equiv -fracpi4 equiv frac3pi4 equiv frac7pi4pmodpi$$
Solving this yields solutions $x=pi n +frac7pi8$ and $x=pi n +frac3pi8$

share|cite|improve this answer

edited Jul 19 at 13:25

answered Jul 19 at 13:18

Shrey Joshi

1389

Hint: $tan(x)$ is an odd function so $tan(-x)=-tan(x)$

Solution:

By above we get $tan(x-pi/4)=tan(-x-pi/2)$ so
$$x-pi/4 equiv -x-pi/2 pmodpi$$
(since the tangent function has a period of $pi$)
$$2x equiv -fracpi4 equiv frac3pi4 equiv frac7pi4pmodpi$$
Solving this yields solutions $x=pi n +frac7pi8$ and $x=pi n +frac3pi8$

share|cite|improve this answer

edited Jul 19 at 13:25

answered Jul 19 at 13:18

Shrey Joshi

1389

share|cite|improve this answer

share|cite|improve this answer

edited Jul 19 at 13:25

edited Jul 19 at 13:25

edited Jul 19 at 13:25

answered Jul 19 at 13:18

Shrey Joshi

1389

answered Jul 19 at 13:18

Shrey Joshi

1389

answered Jul 19 at 13:18

Shrey Joshi

1389

1389

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Hint

$$tan x+tan y=fracsin(x+y)cos xcos y$$

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answered Jul 19 at 13:26

Arnaldo

18k42146

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up vote
0
down vote

Hint

$$tan x+tan y=fracsin(x+y)cos xcos y$$

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answered Jul 19 at 13:26

Arnaldo

18k42146

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up vote
0
down vote

up vote
0
down vote

Hint

$$tan x+tan y=fracsin(x+y)cos xcos y$$

share|cite|improve this answer

answered Jul 19 at 13:26

Arnaldo

18k42146

Hint

$$tan x+tan y=fracsin(x+y)cos xcos y$$

share|cite|improve this answer

answered Jul 19 at 13:26

Arnaldo

18k42146

share|cite|improve this answer

share|cite|improve this answer

answered Jul 19 at 13:26

Arnaldo

18k42146

answered Jul 19 at 13:26

Arnaldo

18k42146

answered Jul 19 at 13:26

Arnaldo

18k42146

18k42146

add a comment | 

add a comment | 

 

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