Advertisements
Advertisements
Question
Write the value of \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
Advertisements
Solution
\[\frac{\sin A + \sin3A}{\cos A + \cos3A}\]
\[ = \frac{2\sin\left( \frac{A + 3A}{2} \right)\cos\left( \frac{A - 3A}{2} \right)}{2\cos\left( \frac{A + 3A}{2} \right)\cos\left( \frac{A - 3A}{2} \right)} \left[ \because \sin A + \sin B = 2\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right), \text{ and }\cos A + \cos B = 2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right]\]
\[ = \frac{\sin2A \cos\left( - A \right)}{\cos2A \cos\left( - A \right)}\]
\[ = \frac{\sin2A \cos A}{\cos2A \cos A}\]
\[ =\tan2A\]
APPEARS IN
RELATED QUESTIONS
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
Prove that:
sin 20° sin 40° sin 60° sin 80° = \[\frac{3}{16}\]
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:
cos 12x + cos 8x
Express each of the following as the product of sines and cosines:
sin 2x + cos 4x
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
cos 55° + cos 65° + cos 175° = 0
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
cos 20° + cos 100° + cos 140° = 0
Prove that:
\[\sin\frac{5\pi}{18} - \cos\frac{4\pi}{9} = \sqrt{3} \sin\frac{\pi}{9}\]
Prove that:
Prove that:
Prove that:
cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A
Prove that:
`sin A + sin 2A + sin 4A + sin 5A = 4 cos (A/2) cos((3A)/2)sin3A`
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos Bcos C
If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.
If A + B = \[\frac{\pi}{3}\] and cos A + cos B = 1, then find the value of cos \[\frac{A - B}{2}\].
If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]
sin 163° cos 347° + sin 73° sin 167° =
The value of sin 50° − sin 70° + sin 10° is equal to
If \[\tan\alpha = \frac{x}{x + 1}\] and
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.
sin A + sin 2A
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:
tan 20° tan 40° tan 80° = `sqrt3`.
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)]`
