Advertisements
Advertisements
Question
If sin A + sin B = α and cos A + cos B = β, then write the value of tan \[\left( \frac{A + B}{2} \right)\].
Advertisements
Solution
Given:
sin A + sin B = α .....(i)
cos A + cos B = β .....(ii)
Dividing (i) by (ii):
\[\Rightarrow \frac{\sin A + \sin B}{\cos A + \cos B} = \frac{\alpha}{\beta}\]
\[ \Rightarrow \frac{2\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)}{2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)} = \frac{\alpha}{\beta} \left[ \because \sin A + \sin B = 2\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)\text{ and }\cos A + \cos B = 2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right]\]
\[ \Rightarrow \frac{\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)}{\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)} = \frac{\alpha}{\beta}\]
\[ \Rightarrow \tan\left( \frac{A + B}{2} \right)=\frac{\alpha}{\beta}\]
APPEARS IN
RELATED QUESTIONS
Prove that:
cos 40° cos 80° cos 160° = \[- \frac{1}{8}\]
Prove that:
tan 20° tan 40° tan 60° tan 80° = 3
Prove that:
sin 20° sin 40° sin 60° sin 80° = \[\frac{3}{16}\]
Express each of the following as the product of sines and cosines:
cos 12x + cos 8x
Prove that:
sin 50° + sin 10° = cos 20°
Prove that:
sin 40° + sin 20° = cos 10°
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
cos 20° + cos 100° + cos 140° = 0
Prove that:
Prove that:
`sin A + sin 2A + sin 4A + sin 5A = 4 cos (A/2) cos((3A)/2)sin3A`
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:
Prove that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), prove that tan A tan B tan C + tan D = 0.
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 + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
The value of cos 52° + cos 68° + cos 172° is
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:
cos 7θ sin 3θ
Express the following as the product of sine and cosine.
cos 2A + cos 4A
Express the following as the product of sine and cosine.
sin 6θ – sin 2θ
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
(cos α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`
Prove that:
sin A sin(60° + A) sin(60° – A) = `1/4` sin 3A
Prove that:
`(cos 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A
Prove that cos 20° cos 40° cos 60° cos 80° = `3/16`.
Evaluate:
sin 50° – sin 70° + sin 10°
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.
If tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:
