Advertisements
Advertisements
प्रश्न
If α + β − γ = π and sin2 α +sin2 β − sin2 γ = λ sin α sin β cos γ, then write the value of λ.
Advertisements
उत्तर
\[\text{ Given }:\]
\[\gamma = - \left[ \pi - (\alpha + \beta) \right]\]
\[\text{ Also }, \]
\[\lambda = \frac{\sin^2 \alpha + \sin^2 \beta - \sin^2 \left[ - (\pi - (\alpha + \beta) \right]}{\sin\alpha \sin\beta \cos( - (\pi - (\alpha + \beta))} \]
\[ = \frac{\sin^2 \alpha + \sin^2 \beta - (\sin(\alpha + \beta) )^2}{- (\sin\alpha \sin\beta\cos(\alpha + \beta))} \left[ \sin \left( \pi - \theta \right) = \sin \theta and \cos\left( \pi - \theta \right) = - \cos \theta \right]\]
\[ = \frac{\sin^2 \alpha + \sin^2 \beta - \sin^2 \alpha \cos^2 \beta - \cos^2 \alpha \sin^2 \beta - 2\sin\alpha \sin\beta \cos\alpha \cos\beta}{- (\sin\alpha \sin\beta \cos\alpha \cos\beta - \sin^2 \alpha \sin^2 \beta)}\]
\[ = \frac{\sin^2 \alpha(1 - \cos^2 \beta) + \sin^2 \beta(1 - \cos^2 \alpha) - 2\sin\alpha \sin\beta \cos\alpha \cos\beta}{\sin^2 \alpha \sin^2 \beta - \sin\alpha \sin\beta \cos\alpha \cos\beta}\]
\[ = \frac{2 \sin^2 \alpha \sin^2 \beta - 2\sin\alpha \sin\beta \cos\alpha \cos\beta}{\sin^2 \alpha \sin^2 \beta - \sin\alpha \sin\beta \cos\alpha \cos\beta}\]
\[ = 2\]
APPEARS IN
संबंधित प्रश्न
Prove the following:
`cos ((3pi)/4 + x) - cos((3pi)/4 - x) = -sqrt2 sin x`
Prove the following:
sin2 6x – sin2 4x = sin 2x sin 10x
Prove the following:
`(sin x - siny)/(cos x + cos y)= tan (x -y)/2`
Prove that: `(cos x + cos y)^2 + (sin x - sin y )^2 = 4 cos^2 (x + y)/2`
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 78° cos 18° − cos 78° sin 18°
Evaluate the following:
cos 47° cos 13° − sin 47° sin 13°
Evaluate the following:
cos 80° cos 20° + sin 80° sin 20°
If \[\cos A = - \frac{12}{13}\text{ and }\cot B = \frac{24}{7}\], where A lies in the second quadrant and B in the third quadrant, find the values of the following:
sin (A + B)
If \[\cos A = - \frac{12}{13}\text{ and }\cot B = \frac{24}{7}\], where A lies in the second quadrant and B in the third quadrant, find the values of the following:
cos (A + B)
Prove that:
Prove that: \[\frac{\sin \left( A + B \right) + \sin \left( A - B \right)}{\cos \left( A + B \right) + \cos \left( A - B \right)} = \tan A\]
Prove that:
\[\frac{\tan \left( A + B \right)}{\cot \left( A - B \right)} = \frac{\tan^2 A - \tan^2 B}{1 - \tan^2 A \tan^2 B}\]
Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1
If x lies in the first quadrant and \[\cos x = \frac{8}{17}\], then prove that:
If tan x + \[\tan \left( x + \frac{\pi}{3} \right) + \tan \left( x + \frac{2\pi}{3} \right) = 3\], then prove that \[\frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x} = 1\].
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 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\[\sin\theta = \frac{\lambda + 1}{\lambda - 1}\sin\phi\]
Find the maximum and minimum values of each of the following trigonometrical expression:
12 sin x − 5 cos x
Reduce each of the following expressions to the sine and cosine of a single expression:
24 cos x + 7 sin x
Write the maximum and minimum values of 3 cos x + 4 sin x + 5.
If a = b \[\cos \frac{2\pi}{3} = c \cos\frac{4\pi}{3}\] then write the value of ab + bc + ca.
If cot (α + β) = 0, sin (α + 2β) is equal to
If sin (π cos x) = cos (π sin x), then sin 2x = ______.
If \[\tan\theta = \frac{1}{2}\] and \[\tan\phi = \frac{1}{3}\], then the value of \[\tan\phi = \frac{1}{3}\] is
The value of cos (36° − A) cos (36° + A) + cos (54° + A) cos (54° − A) is
If A − B = π/4, then (1 + tan A) (1 − tan B) is equal to
If α and β are the solutions of the equation a tan θ + b sec θ = c, then show that tan (α + β) = `(2ac)/(a^2 - c^2)`.
If sin(θ + α) = a and sin(θ + β) = b, then prove that cos 2(α - β) - 4ab cos(α - β) = 1 - 2a2 - 2b2
[Hint: Express cos(α - β) = cos((θ + α) - (θ + β))]
Find the general solution of the equation `(sqrt(3) - 1) costheta + (sqrt(3) + 1) sin theta` = 2
[Hint: Put `sqrt(3) - 1` = r sinα, `sqrt(3) + 1` = r cosα which gives tanα = `tan(pi/4 - pi/6)` α = `pi/12`]
If sinθ + cosecθ = 2, then sin2θ + cosec2θ is equal to ______.
If f(x) = cos2x + sec2x, then ______.
[Hint: A.M ≥ G.M.]
The value of tan3A - tan2A - tanA is equal to ______.
The value of `cot(pi/4 + theta)cot(pi/4 - theta)` is ______.
If sinx + cosx = a, then |sinx – cosx| = ______.
The maximum distance of a point on the graph of the function y = `sqrt(3)` sinx + cosx from x-axis is ______.
State whether the statement is True or False? Also give justification.
If tanθ + tan2θ + `sqrt(3)` tanθ tan2θ = `sqrt(3)`, then θ = `("n"pi)/3 + pi/9`
