मराठी

If Sin X + Sin Y = \[\Sqrt{3}\] (Cos Y − Cos X), Then Sin 3x + Sin 3y = - Mathematics

Advertisements
Advertisements

प्रश्न

If sin x + sin y = \[\sqrt{3}\] (cos y − cos x), then sin 3x + sin 3y =

 

पर्याय

  • 2 sin 3x

  • 0

  • 1

  • none of these

MCQ
बेरीज
Advertisements

उत्तर

We have,
sin x + sin y = \[\sqrt{3}\] (cos y − cos x)
\[\Rightarrow 2\sin\left( \frac{x + y}{2} \right) \cos\left( \frac{x - y}{2} \right) = 2\sqrt{3}\sin\left( \frac{x + y}{2} \right) \sin\left( \frac{x - y}{2} \right)\]
\[ \Rightarrow 2\sin\left( \frac{x + y}{2} \right)\cos\left( \frac{x - y}{2} \right) - 2\sqrt{3}\sin\left( \frac{x + y}{2} \right)\sin\left( \frac{x - y}{2} \right) = 0\]
\[ \Rightarrow 2\sin\left( \frac{x + y}{2} \right)\left[ \cos\left( \frac{x - y}{2} \right) - \sqrt{3}\sin\frac{x - y}{2} \right] = 0\]
\[ \Rightarrow \sin\left( \frac{x + y}{2} \right)\left[ \cos\left( \frac{x - y}{2} \right) - \sqrt{3}\sin\frac{x - y}{2} \right] = 0\]
\[ \Rightarrow \sin\frac{x + y}{2} = 0 \text{ or }, \cos\left( \frac{x - y}{2} \right)-\sqrt{3}\sin\left( \frac{x - y}{2} \right)=0\]
\[\Rightarrow\frac{x + y}{2}=0\text{ or },\tan\left( \frac{x - y}{2} \right)=\frac{1}{\sqrt{3}}=\tan\frac{\pi}{6}\]
\[\Rightarrow x=-y\text{ or },\frac{x - y}{2}=\frac{\pi}{6}\]
\[\Rightarrow x=-y\text{ or },x-y=\frac{\pi}{3}\]

Case - I
Where x = -y

In this case,
sin3x + sin3y = sin(-3y) + sin3y = - sin3y + sin3y = 0
Case - II
Where x - y = `pi/3`
or, \[ 3x = \pi + 3y\]
\[\text{So,} \sin 3x + \sin 3y = \sin\left( \pi + 3y \right) + \sin 3y\]
\[ = - \sin 3y + \sin 3y\]
\[ = 0\]

shaalaa.com
Transformation Formulae
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 8: Transformation formulae - Exercise 8.4 [पृष्ठ २२]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 11
पाठ 8 Transformation formulae
Exercise 8.4 | Q 13 | पृष्ठ २२

संबंधित प्रश्‍न

Show that :

\[\sin 25^\circ \cos 115^\circ = \frac{1}{2}\left( \sin 140^\circ - 1 \right)\]

Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]

 


Prove that:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]

 


Prove that:
tan 20° tan 40° tan 60° tan 80° = 3

 


Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]

 


Show that:
sin A sin (B − C) + sin B sin (C − A) + sin C sin (A − B) = 0


Prove that:
 cos 100° + cos 20° = cos 40°


Prove that:
sin 50° + sin 10° = cos 20°


Prove that:
sin 105° + cos 105° = cos 45°


Prove that:
 sin 50° − sin 70° + sin 10° = 0



Prove that:

\[\cos\left( \frac{3\pi}{4} + x \right) - \cos\left( \frac{3\pi}{4} - x \right) = - \sqrt{2} \sin x\]

 


Prove that:

\[\cos\left( \frac{\pi}{4} + x \right) + \cos\left( \frac{\pi}{4} - x \right) = \sqrt{2} \cos x\]

 


Prove that:
sin 47° + cos 77° = cos 17°


Prove that:
cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A


Prove that:
sin 3A + sin 2A − sin A = 4 sin A cos \[\frac{A}{2}\] \[\frac{3A}{2}\]

 


Prove that:

cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −\[\frac{3}{4}\]

 


Prove that:
\[\sin\frac{x}{2}\sin\frac{7x}{2} + \sin\frac{3x}{2}\sin\frac{11x}{2} = \sin 2x \sin 5x .\]

 


Prove that:

\[\frac{\sin A + \sin 3A}{\cos A - \cos 3A} = \cot A\]

 


Prove that:

\[\frac{\sin A + \sin B}{\sin A - \sin B} = \tan \left( \frac{A + B}{2} \right) \cot \left( \frac{A - B}{2} \right)\]

Prove that:

\[\frac{\sin 11A \sin A + \sin 7A \sin 3A}{\cos 11A \sin A + \cos 7A \sin 3A} = \tan 8A\]

Prove that:

\[\frac{\sin A \sin 2A + \sin 3A \sin 6A}{\sin A \cos 2A + \sin 3A \cos 6A} = \tan 5A\]

Prove that:

\[\frac{\sin \left( \theta + \phi \right) - 2 \sin \theta + \sin \left( \theta - \phi \right)}{\cos \left( \theta + \phi \right) - 2 \cos \theta + \cos \left( \theta - \phi \right)} = \tan \theta\]

Prove that:
cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos Bcos C


\[\text{ If } \cos A + \cos B = \frac{1}{2}\text{ and }\sin A + \sin B = \frac{1}{4},\text{ prove that }\tan\left( \frac{A + B}{2} \right) = \frac{1}{2} .\]

 


\[\text{ If }\frac{\cos (A - B)}{\cos (A + B)} + \frac{\cos (C + D)}{\cos (C - D)} = 0, \text {Prove that }\tan A \tan B \tan C \tan D = - 1\]

 


If sin A + sin B = α and cos A + cos B = β, then write the value of tan \[\left( \frac{A + B}{2} \right)\].

 

The value of cos 52° + cos 68° + cos 172° is


If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=


If A, B, C are in A.P., then \[\frac{\sin A - \sin C}{\cos C - \cos A}\]=

 

If sin (B + C − A), sin (C + A − B), sin (A + B − C) are in A.P., then cot A, cot B and cot Care in


If \[\tan\alpha = \frac{x}{x + 1}\] and 

\[\tan\beta = \frac{1}{2x + 1}\], then
\[\tan\beta = \frac{1}{2x + 1}\] is equal to

 


Express the following as the sum or difference of sine or cosine:

`cos  (7"A")/3 sin  (5"A")/3`


Express the following as the product of sine and cosine.

cos 2θ – cos θ


Prove that:

cos 20° cos 40° cos 80° = `1/8`


Evaluate-

cos 20° + cos 100° + cos 140°


If cos A + cos B = `1/2` and sin A + sin B = `1/4`, prove that tan `(("A + B")/2) = 1/2`


If sin(y + z – x), sin(z + x – y), sin(x + y – z) are in A.P, then prove that tan x, tan y and tan z are in A.P.


Find the value of tan22°30′. `["Hint:"  "Let" θ = 45°, "use" tan  theta/2 = (sin  theta/2)/(cos  theta/2) = (2sin  theta/2 cos  theta/2)/(2cos^2  theta/2) = sintheta/(1 + costheta)]`


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×