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Parametric differentiation

2

$begingroup$

The parametric equations of a curve are
$$begincasesx(t)=e^-tcos t\y(t)=e^-tsin tendcases$$

Show that
$$fracdydx= tanleft(t-fracpi4right)$$

I did the differentiation correct which is

$$fracsin t-cos tcos t+sin t$$

but I don't know how can I reach the final answer? how can this be changed to $tanleft(t-dfracpi4right)$

derivatives parametric

share|cite|improve this question

edited Apr 7 '15 at 23:46

user170231

4,21411429

asked Apr 7 '15 at 23:33

lam97lam97

284

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Do you know the identity for $tan(A-B)$? Try that on $tan(t-pi/4)$ and simplify it to the other expression.
  $endgroup$
  – Rory Daulton
  Apr 7 '15 at 23:37
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Recall the angle difference identities: $$sin(xpm y)=sin xcos ypm cos xsin y\ cos(xpm y)=cos xcos ympsin xsin y$$
  $endgroup$
  – user170231
  Apr 7 '15 at 23:38
  

  
  
  
  

add a comment | 

2

$begingroup$

The parametric equations of a curve are
$$begincasesx(t)=e^-tcos t\y(t)=e^-tsin tendcases$$

Show that
$$fracdydx= tanleft(t-fracpi4right)$$

I did the differentiation correct which is

$$fracsin t-cos tcos t+sin t$$

but I don't know how can I reach the final answer? how can this be changed to $tanleft(t-dfracpi4right)$

derivatives parametric

share|cite|improve this question

edited Apr 7 '15 at 23:46

user170231

4,21411429

asked Apr 7 '15 at 23:33

lam97lam97

284

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Do you know the identity for $tan(A-B)$? Try that on $tan(t-pi/4)$ and simplify it to the other expression.
  $endgroup$
  – Rory Daulton
  Apr 7 '15 at 23:37
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Recall the angle difference identities: $$sin(xpm y)=sin xcos ypm cos xsin y\ cos(xpm y)=cos xcos ympsin xsin y$$
  $endgroup$
  – user170231
  Apr 7 '15 at 23:38
  

  
  
  
  

add a comment | 

$begingroup$

The parametric equations of a curve are
$$begincasesx(t)=e^-tcos t\y(t)=e^-tsin tendcases$$

Show that
$$fracdydx= tanleft(t-fracpi4right)$$

I did the differentiation correct which is

$$fracsin t-cos tcos t+sin t$$

but I don't know how can I reach the final answer? how can this be changed to $tanleft(t-dfracpi4right)$

derivatives parametric

share|cite|improve this question

edited Apr 7 '15 at 23:46

user170231

4,21411429

asked Apr 7 '15 at 23:33

lam97lam97

284

$endgroup$

share|cite|improve this question

edited Apr 7 '15 at 23:46

user170231

4,21411429

asked Apr 7 '15 at 23:33

lam97lam97

284

share|cite|improve this question

edited Apr 7 '15 at 23:46

user170231

4,21411429

asked Apr 7 '15 at 23:33

lam97lam97

284

edited Apr 7 '15 at 23:46

user170231

4,21411429

edited Apr 7 '15 at 23:46

user170231

4,21411429

asked Apr 7 '15 at 23:33

lam97lam97

284

asked Apr 7 '15 at 23:33

lam97lam97

284

2 Answers
2

active

oldest

votes

1

$begingroup$

$$fracsin t-cos tcos t+sin t=fracsqrt 2(sin t,cos fracpi 4-sinfracpi 4,cos t)sqrt 2(cos t,cos fracpi 4+sin t,sinfracpi 4)=fracsin(t-fracpi 4)cos(t-fracpi 4).$$

share|cite|improve this answer

answered Apr 7 '15 at 23:42

BernardBernard

124k741117

$endgroup$

add a comment | 

1

$begingroup$

Use the formula $tan(a-b)$:

$$tan(a-b) = fractan(a) -tan(b)1 + tan(a)tan(b)$$

Substituting this in, we get:

$$tan(t-pi/4) = fractan(t) -tanleft(fracpi4right)1 + tan(t)tanleft(fracpi4right) = fractan(t) - 11 + tan(t) =fraccfracsin(t) -cos(t)cos(t)cfracsin(t) +cos(t)cos(t) = fracsin(t) - cos(t)sin(t) + cos(t)$$

This equals that initial answer you arrived at when differentiating the parametric equation.

share|cite|improve this answer

edited Apr 8 at 18:30

Bernard

124k741117

answered Apr 7 '15 at 23:43

Varun IyerVarun Iyer

5,362926

$endgroup$

add a comment | 

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1

$begingroup$

$$fracsin t-cos tcos t+sin t=fracsqrt 2(sin t,cos fracpi 4-sinfracpi 4,cos t)sqrt 2(cos t,cos fracpi 4+sin t,sinfracpi 4)=fracsin(t-fracpi 4)cos(t-fracpi 4).$$

share|cite|improve this answer

answered Apr 7 '15 at 23:42

BernardBernard

124k741117

$endgroup$

add a comment | 

1

$begingroup$

$$fracsin t-cos tcos t+sin t=fracsqrt 2(sin t,cos fracpi 4-sinfracpi 4,cos t)sqrt 2(cos t,cos fracpi 4+sin t,sinfracpi 4)=fracsin(t-fracpi 4)cos(t-fracpi 4).$$

share|cite|improve this answer

answered Apr 7 '15 at 23:42

BernardBernard

124k741117

$endgroup$

add a comment | 

$begingroup$

$$fracsin t-cos tcos t+sin t=fracsqrt 2(sin t,cos fracpi 4-sinfracpi 4,cos t)sqrt 2(cos t,cos fracpi 4+sin t,sinfracpi 4)=fracsin(t-fracpi 4)cos(t-fracpi 4).$$

share|cite|improve this answer

answered Apr 7 '15 at 23:42

BernardBernard

124k741117

$endgroup$

share|cite|improve this answer

answered Apr 7 '15 at 23:42

BernardBernard

124k741117

answered Apr 7 '15 at 23:42

BernardBernard

124k741117

answered Apr 7 '15 at 23:42

BernardBernard

124k741117

1

$begingroup$

Use the formula $tan(a-b)$:

$$tan(a-b) = fractan(a) -tan(b)1 + tan(a)tan(b)$$

Substituting this in, we get:

$$tan(t-pi/4) = fractan(t) -tanleft(fracpi4right)1 + tan(t)tanleft(fracpi4right) = fractan(t) - 11 + tan(t) =fraccfracsin(t) -cos(t)cos(t)cfracsin(t) +cos(t)cos(t) = fracsin(t) - cos(t)sin(t) + cos(t)$$

This equals that initial answer you arrived at when differentiating the parametric equation.

share|cite|improve this answer

edited Apr 8 at 18:30

Bernard

124k741117

answered Apr 7 '15 at 23:43

Varun IyerVarun Iyer

5,362926

$endgroup$

add a comment | 

1

$begingroup$

Use the formula $tan(a-b)$:

$$tan(a-b) = fractan(a) -tan(b)1 + tan(a)tan(b)$$

Substituting this in, we get:

$$tan(t-pi/4) = fractan(t) -tanleft(fracpi4right)1 + tan(t)tanleft(fracpi4right) = fractan(t) - 11 + tan(t) =fraccfracsin(t) -cos(t)cos(t)cfracsin(t) +cos(t)cos(t) = fracsin(t) - cos(t)sin(t) + cos(t)$$

This equals that initial answer you arrived at when differentiating the parametric equation.

share|cite|improve this answer

edited Apr 8 at 18:30

Bernard

124k741117

answered Apr 7 '15 at 23:43

Varun IyerVarun Iyer

5,362926

$endgroup$

add a comment | 

$begingroup$

Use the formula $tan(a-b)$:

$$tan(a-b) = fractan(a) -tan(b)1 + tan(a)tan(b)$$

Substituting this in, we get:

$$tan(t-pi/4) = fractan(t) -tanleft(fracpi4right)1 + tan(t)tanleft(fracpi4right) = fractan(t) - 11 + tan(t) =fraccfracsin(t) -cos(t)cos(t)cfracsin(t) +cos(t)cos(t) = fracsin(t) - cos(t)sin(t) + cos(t)$$

This equals that initial answer you arrived at when differentiating the parametric equation.

share|cite|improve this answer

edited Apr 8 at 18:30

Bernard

124k741117

answered Apr 7 '15 at 23:43

Varun IyerVarun Iyer

5,362926

$endgroup$

share|cite|improve this answer

edited Apr 8 at 18:30

Bernard

124k741117

answered Apr 7 '15 at 23:43

Varun IyerVarun Iyer

5,362926

edited Apr 8 at 18:30

Bernard

124k741117

edited Apr 8 at 18:30

Bernard

124k741117

answered Apr 7 '15 at 23:43

Varun IyerVarun Iyer

5,362926

answered Apr 7 '15 at 23:43

Varun IyerVarun Iyer

5,362926