Advertisements
Advertisements
प्रश्न
If sin α + sin β = a and cos α + cos β = b, show that
Advertisements
उत्तर
\[a^2 + b^2 = \left( \sin\alpha + \sin\beta \right)^2 + (\cos\alpha + \cos\beta)^2 \]
\[ = \sin^2 \alpha + \sin^2 \beta + \cos^2 \alpha + \cos^2 \beta + 2\sin\alpha\sin\beta + 2\cos\alpha\cos\beta\]
\[ = 2 + 2 \cos(\alpha - \beta)\] ........(1)
Now,
\[ \Rightarrow b^2 - a^2 = {(\cos\alpha + \cos\beta)}^2 - \left( \sin\alpha + \sin\beta \right)^2 \]
\[ \Rightarrow b^2 - a^2 = \cos^2 \alpha + \cos^2 \beta - \sin^2 \alpha - \sin^2 \beta + 2\cos\alpha\cos\beta - 2\sin\alpha\sin\beta\]
\[ \Rightarrow b^2 - a^2 = ( \cos^2 \alpha - \sin^2 \beta) + ( \cos^2 \beta - \sin^2 \alpha) - 2\cos(\alpha + \beta)\]
\[ \Rightarrow b^2 - a^2 = 2\cos(\alpha + \beta)\cos(\alpha - \beta) + 2\cos(\alpha - \beta)\]
\[ \Rightarrow b^2 - a^2 = \cos(\alpha + \beta)(2 + 2 \cos(\alpha - \beta)) \] .........(2)
From (1) and (2), we have
\[ \Rightarrow b^2 - a^2 = \cos(\alpha + \beta)\left( a^2 + b^2 \right) \]
\[\frac{b^2 - a^2}{a^2 + b^2} = \cos(\alpha + \beta)\]
APPEARS IN
संबंधित प्रश्न
Prove that `2 sin^2 pi/6 + cosec^2 (7pi)/6 cos^2 pi/3 = 3/2`
Prove that: `2 sin^2 (3pi)/4 + 2 cos^2 pi/4 + 2 sec^2 pi/3 = 10`
Prove the following:
sin 2x + 2sin 4x + sin 6x = 4cos2 x sin 4x
Prove the following:
`(cos9x - cos5x)/(sin17x - sin 3x) = - (sin2x)/(cos 10x)`
Prove the following:
`(sin x - siny)/(cos x + cos y)= tan (x -y)/2`
Prove the following:
`(sin x + sin 3x)/(cos x + cos 3x) = tan 2x`
Prove the following:
`tan 4x = (4tan x(1 - tan^2 x))/(1 - 6tan^2 x + tan^4 x)`
If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
sin (A − B)
If \[\sin A = \frac{12}{13}\text{ and } \sin B = \frac{4}{5}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
cos (A + B)
If \[\cos A = - \frac{24}{25}\text{ and }\cos B = \frac{3}{5}\], where π < A < \[\frac{3\pi}{2}\text{ and }\frac{3\pi}{2}\]< B < 2π, find the following:
cos (A + B)
If \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
tan (A - B)
Evaluate the following:
sin 36° cos 9° + cos 36° sin 9°
Evaluate the following:
cos 80° cos 20° + sin 80° sin 20°
Prove that
Prove that:
Prove that:
Prove that:
sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin A.
Prove that:
tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
Prove that:
tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x
Prove that sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin A.
If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.
If sin (α + β) = 1 and sin (α − β) \[= \frac{1}{2}\], where 0 ≤ α, \[\beta \leq \frac{\pi}{2}\], then find the values of tan (α + 2β) and tan (2α + β).
If sin α + sin β = a and cos α + cos β = b, show that
If angle \[\theta\] is divided into two parts such that the tangents of one part is \[\lambda\] times the tangent of other, and \[\phi\] is their difference, then show that\[\sin\theta = \frac{\lambda + 1}{\lambda - 1}\sin\phi\]
If \[\tan\theta = \frac{\sin\alpha - \cos\alpha}{\sin\alpha + \cos\alpha}\] , then show that \[\sin\alpha + \cos\alpha = \sqrt{2}\cos\theta\].
Find the maximum and minimum values of each of the following trigonometrical expression:
12 sin x − 5 cos x
Show that sin 100° − sin 10° is positive.
Write the maximum and minimum values of 3 cos x + 4 sin x + 5.
If cos (A − B) \[= \frac{3}{5}\] and tan A tan B = 2, then
If \[\tan\alpha = \frac{x}{x + 1}\] and \[\tan\alpha = \frac{x}{x + 1}\], then \[\alpha + \beta\] is equal to
Express the following as the sum or difference of sines and cosines:
2 cos 3x sin 2xa
Express the following as the sum or difference of sines and cosines:
2 cos 7x cos 3x
Show that 2 sin2β + 4 cos (α + β) sin α sin β + cos 2(α + β) = cos 2α
Match each item given under column C1 to its correct answer given under column C2.
| C1 | C2 |
| (a) `(1 - cosx)/sinx` | (i) `cot^2 x/2` |
| (b) `(1 + cosx)/(1 - cosx)` | (ii) `cot x/2` |
| (c) `(1 + cosx)/sinx` | (iii) `|cos x + sin x|` |
| (d) `sqrt(1 + sin 2x)` | (iv) `tan x/2` |
The value of sin(45° + θ) - cos(45° - θ) is ______.
If tanA = `1/2`, tanB = `1/3`, then tan(2A + B) is equal to ______.
If α + β = `pi/4`, then the value of (1 + tan α)(1 + tan β) is ______.
If tanθ = `a/b`, then bcos2θ + asin2θ is equal to ______.
In the following match each item given under the column C1 to its correct answer given under the column C2:
| Column A | Column B |
| (a) sin(x + y) sin(x – y) | (i) cos2x – sin2y |
| (b) cos (x + y) cos (x – y) | (ii) `(1 - tan theta)/(1 + tan theta)` |
| (c) `cot(pi/4 + theta)` | (iii) `(1 + tan theta)/(1 - tan theta)` |
| (d) `tan(pi/4 + theta)` | (iv) sin2x – sin2y |
