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Grade 12Trigonometry

If 3sinx + 4cosx = 5, then 4sinx - 3cosx =....???

  1. 0
  2. -5
  3. 1/5
  4. 5

plz explain me in detail... how to do questions like this...

Profile image of Abhijat Pandey
14 Years agoGrade 12
Answers icon

9 Answers

Profile image of KARAN ACHARYA
ApprovedApproved Tutor Answer14 Years ago

ANS    3sinx+4cosx=5

     => 3/5 sinx +4/5 cosx = 1

     let cosA=3/5   => sinA=4/5

     =>  cosAsinx + sinAcosx = 1

     =>  sin(x+A) = 1

Now,

       4sinx - 3cosx

  =   5(4/5sinx - 3/5 cosx)     [multipying numerator and denominator by 5]

  =   5(sinAsinx - cosAcosx)

  =   -5{cos(x+A)} = -5[root{1-(sin(x+A)^2)}] = -5 x 0 = 0          Ans 0

 

TRICK: if asinx+bcosx is given then multiply numerator and denominator by root(a^2 +b^2)

            this method is useful  in many questions 

 

 

 PLEASE       APPROVE       MY       ANSWER       IF       YOU       LIKE       IT

 

Profile image of BARATH P
13 Years ago

3sinx+4cosx=5

Squaring on both sides.

(3sinx+4cosx) 2 = 25

9sin2x +16 cos2x+24sinxcosx=25

9(1- cos2x)+16(1- sin2x)+ 24sinxcosx=25

9-9 cos2x+16-16 sin2x+24sinxcosx=25

25-9 cos2x-16 sin2x+24sinxcosx=25

9 cos2x+16 sin2x-24sinxcosx=0

(3cosx-4sinx) 2 = 0

4sinx-3cosx=0

 

Profile image of Chandrabhushan Reddy Chigarapalli
11 Years ago
3sinx+4cosx=5.
Do sqaring on both sides.
The equation will be as :- 9sin2x+16cos2x+24sinx.cosx=25.
Let the above equation be no.1
Think that the value of 4sinx-3cosx=p.
Do squaring on both sides.
The eqation will be as :- 16sin2x+9cos2x-24sinx.cosx=p2.
Let the above equation be no.2
Add equations no.1 and no.2
You will get as :- 25sin2x+25cos2x=25+p2
25(sin2x+cos2x)=25+p2
25=25+p2                                   (because sin2x+cos2x=1)
p2=0
p=0.
4sinx-3cosx=0.
Therefore answer is 0.
 
 
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Profile image of sumanth
11 Years ago
 

3sinx+4cosx=5

 

Squaring on both sides.

 

(3sinx+4cosx) 2 = 25

 

9sin2x +16 cos2x+24sinxcosx=25

 

9(1- cos2x)+16(1- sin2x)+ 24sinxcosx=25

 

9-9 cos2x+16-16 sin2x+24sinxcosx=25

 

25-9 cos2x-16 sin2x+24sinxcosx=25

 

9 cos2x+16 sin2x-24sinxcosx=0

 

(3cosx-4sinx) 2 = 0

 

4sinx-3cosx=0

Profile image of Lab Bhattacharjee
11 Years ago
HINT:  (3sinx +4cosx)^2+(3cosx-4sinx)^2=(3^2+4^2)(cos^2x+sin^2x)
Profile image of ritesh
10 Years ago
easiest method....given 3sinx+4cosx=5..then find 4sinx-3cosx=?
a=3  b=4  c=5...let our ans to be k
a^2+b^2=c^2+k^2
after putting values nd solving we get...k=0...easy method...
 
 
Profile image of Kushagra Madhukar
6 Years ago
Hello student
This question can be solved using two methods both are quite easy
 
Method-1
Given, 
3sinx + 4cosx = 5
Squaring both sides,
9sin2x + 16cos2x + 24sinxcosx = 25
9sin2x + 16 cos2x + 24sinxcosx = 25(sin2x + cos2x)       [ 25 = 25 x1 = 25 x (sin2x + cos2x)
24sinxcosx = 16sin2x + 9 cos2x
or, (4sinx)2 + (3cosx)2 – 2(4sinx)(3cosx) = 0
or, [4sinx – 3cosx]2 = 0
hence, 4sinx – 3cosx =0
 
Method-2
3sinx + 4cosx = 5
as we can notice- 3, 4, 5 are pythagorean triplet therefore are a part of the same right angled triangle.
Dividing both sides by 5
3/5 sinx + 4/5 cosx = 1
If we take 3/5 = cosA ; then sinA = 4/5
or, sinx cosA + cosx sinA = 1
or, sin(x + A) = 1
 
therefore,
cos(x + A) = [1 – {sin(x + A)}2]1/2  = [ 1 – 1 ]½  = 0
cosx cosA – sinx sinA = 0
or, 3/5 cosx – 4/5 sinx = 0
multiplying both sides by -5, we get,
4sinx – 3cosx = 0
 
You can use either of these methods in questions like this. But the 2nd method might be easier in case of some problems.
Hope it helps.
Regards,
Kushagra
Profile image of Vikas TU
6 Years ago
Dear student 
Question is not clear 
Please upload an image.
We will happy to help you.
Good luck 
Cheers 
 
Profile image of Zaid
5 Years ago
3sinx + 4cosx = 5
as we can notice- 3, 4, 5 are pythagorean triplet therefore are a part of the same right angled triangle.
Dividing both sides by 5
3/5 sinx + 4/5 cosx = 1
If we take 3/5 = cosA ; then sinA = 4/5
or, sinx cosA + cosx sinA = 1
or, sin(x + A) = 1
 
therefore,
cos(x + A) = [1 – {sin(x + A)}2]1/2  = [ 1 – 1 ]½  = 0
cosx cosA – sinx sinA = 0
or, 3/5 cosx – 4/5 sinx = 0
multiplying both sides by -5, we get,
4sinx – 3cosx = 0

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