Advertisements
Advertisements
Question
Prove that:
`sin A + sin 2A + sin 4A + sin 5A = 4 cos (A/2) cos((3A)/2)sin3A`
Advertisements
Solution
Consider LHS:
\[ \sin A + \sin 2A + \sin 4A + \sin 5A\]
\[ = 2\sin \left( \frac{A + 2A}{2} \right) \cos \left( \frac{A - 2A}{2} \right) + 2\sin \left( \frac{4A + 5A}{2} \right) \cos \left( \frac{4A - 5A}{2} \right) \left\{ \because \sin A + \sin B = 2\sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \right\}\]
\[ = 2\sin \left( \frac{3}{2}A \right) \cos \left( - \frac{A}{2} \right) + 2\sin \left( \frac{9}{2}A \right) \cos \left( - \frac{A}{2} \right)\]
\[= 2\sin \left( \frac{3}{2}A \right) \cos \left( \frac{A}{2} \right) + 2\sin \left( \frac{9}{2}A \right) \cos \left( \frac{A}{2} \right)\]
\[ = 2\cos \left( \frac{A}{2} \right)\left\{ \sin \frac{3}{2}A + \sin \frac{9}{2}A \right\}\]
\[ = 2\cos \left( \frac{A}{2} \right) \times 2\sin \left( \frac{\frac{3}{2}A + \frac{9}{2}A}{2} \right) \cos \left( \frac{\frac{3}{2}A - \frac{9}{2}A}{2} \right)\]
\[ = 4\cos \left( \frac{A}{2} \right) \sin 3A \cos \left( - \frac{3}{2}A \right)\]
\[ = 4\cos \frac{A}{2} \cos \left( \frac{3A}{2} \right) \sin 3A\]
= RHS
Hence, LHS = RHS
APPEARS IN
RELATED QUESTIONS
Prove that:
Show that :
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
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 (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
Express each of the following as the product of sines and cosines:
sin 5x − sin x
Express each of the following as the product of sines and cosines:
sin 2x + cos 4x
Prove that:
sin 38° + sin 22° = sin 82°
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
cos 20° + cos 100° + cos 140° = 0
Prove that:
Prove that:
cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −\[\frac{3}{4}\]
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
If \[x \cos\theta = y \cos\left( \theta + \frac{2\pi}{3} \right) = z \cos\left( \theta + \frac{4\pi}{3} \right)\], prove that \[xy + yz + zx = 0\]
If cos A = m cos B, then write the value of \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]
sin 163° cos 347° + sin 73° sin 167° =
If sin 2 θ + sin 2 ϕ = \[\frac{1}{2}\] and cos 2 θ + cos 2 ϕ = \[\frac{3}{2}\], then cos2 (θ − ϕ) =
cos 35° + cos 85° + cos 155° =
If A, B, C are in A.P., then \[\frac{\sin A - \sin C}{\cos C - \cos A}\]=
If sin x + sin y = \[\sqrt{3}\] (cos y − cos x), then sin 3x + sin 3y =
Express the following as the sum or difference of sine or cosine:
`sin "A"/8 sin (3"A")/8`
Express the following as the sum or difference of sine or cosine:
cos 7θ sin 3θ
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
tan 20° tan 40° tan 80° = `sqrt3`.
Prove that:
sin A sin(60° + A) sin(60° – A) = `1/4` sin 3A
Prove that:
`(cos 2"A" - cos 3"A")/(sin "2A" + sin "3A") = tan "A"/2`
Prove that:
`(cos 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A
