मराठी

If sin α + sin β = a and cos α − cos β = b then tan α − β 2 = - Mathematics

Advertisements
Advertisements

प्रश्न

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

 

पर्याय

  • \[- \frac{a}{b}\]

     

  • \[- \frac{b}{a}\]

     

  • \[\sqrt{a^2 + b^2}\]

     

  • none of these

MCQ
Advertisements

उत्तर

\[- \frac{b}{a}\]

\[\text{ Given } : \]
\[sin\alpha + sin\beta = a\]
\[ \Rightarrow 2\sin\frac{\alpha + \beta}{2}\cos\frac{\alpha - \beta}{2} = a . . . (1)\]
\[\text{ Also } , \]
\[cos\alpha + cos\beta = b\]
\[ \Rightarrow - 2\sin\frac{\alpha + \beta}{2}\sin\frac{\alpha - \beta}{2} = b . . . (2)\]
\[\text{ On dividing (1) by (2), we get} \]
\[\frac{- \cos\frac{\alpha - \beta}{2}}{\sin\frac{\alpha - \beta}{2}} = \frac{a}{b}\]
\[ \Rightarrow \frac{- \sin\frac{\alpha - \beta}{2}}{\cos\frac{\alpha - \beta}{2}} = \frac{b}{a}\]
\[ \Rightarrow \tan\frac{\alpha - \beta}{2} = - \frac{b}{a}\]

shaalaa.com
Values of Trigonometric Functions at Multiples and Submultiples of an Angle
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 9: Values of Trigonometric function at multiples and submultiples of an angle - Exercise 9.5 [पृष्ठ ४३]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 11
पाठ 9 Values of Trigonometric function at multiples and submultiples of an angle
Exercise 9.5 | Q 11 | पृष्ठ ४३

संबंधित प्रश्‍न

Prove that:  \[\frac{\sin 2x}{1 - \cos 2x} = cot x\]


Prove that:  \[\frac{\sin x + \sin 2x}{1 + \cos x + \cos 2x} = \tan x\]

 

Prove that: \[\cos^3 2x + 3 \cos 2x = 4\left( \cos^6 x - \sin^6 x \right)\]


Prove that: \[\cos^2 \left( \frac{\pi}{4} - x \right) - \sin^2 \left( \frac{\pi}{4} - x \right) = \sin 2x\]


Prove that: \[\cos^6 A - \sin^6 A = \cos 2A\left( 1 - \frac{1}{4} \sin^2 2A \right)\]

 

Prove that: \[\cot^2 x - \tan^2 x = 4 \cot 2 x  \text{ cosec }  2 x\]

 

If  \[\sin x = \frac{\sqrt{5}}{3}\] and x lies in IInd quadrant, find the values of \[\cos\frac{x}{2}, \sin\frac{x}{2} \text{ and }  \tan \frac{x}{2}\] . 

 

 


If \[\tan A = \frac{1}{7}\]  and \[\tan B = \frac{1}{3}\] , show that cos 2A = sin 4

 

 


Prove that: \[\cos \frac{\pi}{65} \cos \frac{2\pi}{65} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65} = \frac{1}{64}\]

 

If \[2 \tan \alpha = 3 \tan \beta,\]  prove that \[\tan \left( \alpha - \beta \right) = \frac{\sin 2\beta}{5 - \cos 2\beta}\] .

 

If \[\cos x = \frac{\cos \alpha + \cos \beta}{1 + \cos \alpha \cos \beta}\] , prove that \[\tan\frac{x}{2} = \pm \tan\frac{\alpha}{2}\tan\frac{\beta}{2}\]

 

If \[a \cos2x + b \sin2x = c\]  has α and β as its roots, then prove that

(iii)\[\tan\left( \alpha + \beta \right) = \frac{b}{a}\] 

 


Prove that:  \[\sin 5x = 5 \sin x - 20 \sin^3 x + 16 \sin^5 x\]

 

\[\cot x + \cot\left( \frac{\pi}{3} + x \right) + \cot\left( \frac{\pi}{3} - x \right) = 3 \cot 3x\]

 


Prove that \[\left| \sin x \sin \left( \frac{\pi}{3} - x \right) \sin \left( \frac{\pi}{3} + x \right) \right| \leq \frac{1}{4}\]  for all values of x

 
 

Prove that: \[\sin^2 \frac{2\pi}{5} - \sin^{2 -} \frac{\pi}{3} = \frac{\sqrt{5} - 1}{8}\]

  

Prove that:  \[\cos 78°  \cos 42°  \cos 36° = \frac{1}{8}\]


Prove that: \[\cos 6° \cos 42°   \cos 66°    \cos 78° = \frac{1}{16}\]

 

Prove that: \[\sin\frac{\pi}{5}\sin\frac{2\pi}{5}\sin\frac{3\pi}{5}\sin\frac{4\pi}{5} = \frac{5}{16}\]

 

If  \[\text{ sin } x + \text{ cos } x = a\], then find the value of

\[\sin^6 x + \cos^6 x\] .
 

 


\[8 \sin\frac{x}{8} \cos \frac{x}{2}\cos\frac{x}{4} \cos\frac{x}{8}\]  is equal to 

 


If \[\cos x = \frac{1}{2} \left( a + \frac{1}{a} \right),\]  and \[\cos 3 x = \lambda \left( a^3 + \frac{1}{a^3} \right)\] then \[\lambda =\]

 

 


If  \[2 \tan \alpha = 3 \tan \beta, \text{ then }  \tan \left( \alpha - \beta \right) =\]

 


The value of \[\tan x \sin \left( \frac{\pi}{2} + x \right) \cos \left( \frac{\pi}{2} - x \right)\]

 

The value of \[\frac{\cos 3x}{2 \cos 2x - 1}\]  is equal to

   

If \[\tan \left( \pi/4 + x \right) + \tan \left( \pi/4 - x \right) = \lambda \sec 2x, \text{ then } \]


\[2 \left( 1 - 2 \sin^2 7x \right) \sin 3x\]  is equal to


The value of \[\frac{\sin 5 \alpha - \sin 3\alpha}{\cos 5 \alpha + 2 \cos 4\alpha + \cos 3\alpha} =\]

 

\[\frac{\sin 5x}{\sin x}\]  is equal to

 


If \[\text{ tan } x = \frac{a}{b}\], then \[b \cos 2x + a \sin 2x\]

 

 


The value of sin 20° sin 40° sin 60° sin 80° is ______.


The value of `cos  pi/5 cos  (2pi)/5 cos  (4pi)/5 cos  (8pi)/5`  is ______.


Prove that sin 4A = 4sinA cos3A – 4 cosA sin3A


The value of `(1 - tan^2 15^circ)/(1 + tan^2 15^circ)` is ______.


The value of cos12° + cos84° + cos156° + cos132° is ______.


The value of sin50° – sin70° + sin10° is equal to ______.


The value of `(sin 50^circ)/(sin 130^circ)` is ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×