English

The Value of Cos 52° + Cos 68° + Cos 172° is

Advertisements
Advertisements

Question

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

Options

  • 0

  • 1

  • 2

  • `3/2`

MCQ
Advertisements

Solution

0
\[\cos52^\circ + \cos68^\circ + \cos172^\circ\]
\[ = 2\cos\left( \frac{52^\circ + 68^\circ}{2} \right)\cos\left( \frac{52^\circ - 68^\circ}{2} \right) + \cos172^\circ \left[ \because \cos A + \cos B = 2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right]\]
\[ = 2\cos60^\circ\cos\left( - 8^\circ \right) + \cos172^\circ\]
\[ = 2 \times \frac{1}{2}\cos8^\circ + \cos172^\circ\]
\[ = \cos8^\circ + \cos172^\circ\]
\[ = 2\cos\left( \frac{8^\circ + 172^\circ}{2} \right)\cos\left( \frac{8^\circ - 172^\circ}{2} \right)\]
\[ = 2\cos90^\circ\cos82^\circ\]
\[ = 0\]

shaalaa.com
Transformation Formulae
  Is there an error in this question or solution?
Chapter 8: Transformation formulae - Exercise 8.4 [Page 21]

APPEARS IN

R.D. Sharma Mathematics [English] Class 11
Chapter 8 Transformation formulae
Exercise 8.4 | Q 4 | Page 21

RELATED QUESTIONS

Prove that:

\[2\cos\frac{5\pi}{12}\cos\frac{\pi}{12} = \frac{1}{2}\]

Prove that tan 20° tan 30° tan 40° tan 80° = 1.


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


Prove that:
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]


Express each of the following as the product of sines and cosines:
sin 12x + sin 4x


Prove that:
sin 38° + sin 22° = sin 82°


Prove that:
 cos 55° + cos 65° + cos 175° = 0


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



Prove that:

sin 51° + cos 81° = cos 21°

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

 


Prove that:

\[\frac{\sin 9A - \sin 7A}{\cos 7A - \cos 9A} = \cot 8A\]

Prove that:

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

 


Prove that:

\[\frac{\sin 3A + \sin 5A + \sin 7A + \sin 9A}{\cos 3A + \cos 5A + \cos 7A + \cos 9A} = \tan 6A\]

Prove that:

\[\frac{\sin 5A - \sin 7A + \sin 8A - \sin 4A}{\cos 4A + \cos 7A - \cos 5A - \cos 8A} = \cot 6A\]

Prove that:

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

Prove that:

\[\frac{\sin A + 2 \sin 3A + \sin 5A}{\sin 3A + 2 \sin 5A + \sin 7A} = \frac{\sin 3A}{\sin 5A}\]

Prove that:

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

\[\text{ If }\sin 2A = \lambda \sin 2B, \text{ prove that }\frac{\tan (A + B)}{\tan (A - B)} = \frac{\lambda + 1}{\lambda - 1}\]

 


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

 


If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.

 

If (cos α + cos β)2 + (sin α + sin β)2 = \[\lambda \cos^2 \left( \frac{\alpha - \beta}{2} \right)\], write the value of λ. 


Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].


If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]


If cos A = m cos B, then \[\cot\frac{A + B}{2} \cot\frac{B - A}{2}\]=

 

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

 

Express the following as the product of sine and cosine.

cos 2A + cos 4A


Express the following as the product of sine and cosine.

cos 2θ – cos θ


Prove that:

cos 20° cos 40° cos 80° = `1/8`


Prove that:

sin (A – B) sin C + sin (B – C) sin A + sin(C – A) sin B = 0


If cosec A + sec A = cosec B + sec B prove that cot`(("A + B"))/2` = tan A tan B.


If tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:


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×