Advertisements
Advertisements
प्रश्न
If x cos θ = y cos \[\left( \theta + \frac{2\pi}{3} \right) = z \cos \left( \theta + \frac{4\pi}{3} \right)\]then write the value of \[\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\]
Advertisements
उत्तर
\[\text{ Given }: \]
\[x \cos\theta = y\left( \cos\theta\cos\frac{2\pi}{3} - \sin\theta \sin\frac{2\pi}{3} \right) = z\left( \cos\theta\cos\frac{4\pi}{3} - \sin\theta \sin\frac{4\pi}{3} \right)\]
\[ \Rightarrow x\cos\theta = y\left( - \frac{1}{2}\cos\theta - \frac{\sqrt{3}}{2}\sin\theta \right) = z\left( - \frac{1}{2}\cos\theta + \frac{\sqrt{3}}{2}\sin\theta \right) \]
\[ \Rightarrow x = \frac{y}{2}\left( - 1 - \sqrt{3}\tan\theta \right) = \frac{z}{2}\left( - 1 + \sqrt{3}\tan\theta \right)\]
\[x = \frac{y}{2}\left( - 1 - \sqrt{3}\tan\theta \right)\]
\[z = \frac{y\left( - 1 - \sqrt{3}\tan\theta \right)}{\left( - 1 + \sqrt{3}\tan\theta \right)}\]
\[\text{ Now }, \]
\[\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{2}{y\left( - 1 - \sqrt{3}\tan\theta \right)} + \frac{1}{y} + \frac{\left( - 1 + \sqrt{3}\tan\theta \right)}{y\left( - 1 - \sqrt{3}\tan\theta \right)}\]
\[ = \frac{2 + \left( - 1 - \sqrt{3}\tan\theta \right) + \left( - 1 + \sqrt{3}\tan\theta \right)}{y\left( - 1 - \sqrt{3}\tan\theta \right)}\]
\[ = 0\]
APPEARS IN
संबंधित प्रश्न
Prove that: `sin^2 pi/6 + cos^2 pi/3 - tan^2 pi/4 = -1/2`
Prove that `cot^2 pi/6 + cosec (5pi)/6 + 3 tan^2 pi/6 = 6`
Find the value of: tan 15°
Prove the following: `(tan(pi/4 + x))/(tan(pi/4 - x)) = ((1+ tan x)/(1- tan x))^2`
Prove the following:
`cos ((3pi)/ 2 + x ) cos(2pi + x) [cot ((3pi)/2 - x) + cot (2pi + x)]= 1`
Prove the following:
`cos ((3pi)/4 + x) - cos((3pi)/4 - x) = -sqrt2 sin x`
Prove the following:
`(cos9x - cos5x)/(sin17x - sin 3x) = - (sin2x)/(cos 10x)`
Prove the following:
cos 6x = 32 cos6 x – 48 cos4 x + 18 cos2 x – 1
Prove that: `((sin 7x + sin 5x) + (sin 9x + sin 3x))/((cos 7x + cos 5x) + (cos 9x + cos 3x)) = tan 6x`
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:
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:
sin (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)
Evaluate the following:
cos 47° cos 13° − sin 47° sin 13°
Prove that:
\[\cos^2 45^\circ - \sin^2 15^\circ = \frac{\sqrt{3}}{4}\]
Prove that:
Prove that:
tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x
If tan A = x tan B, prove that
\[\frac{\sin \left( A - B \right)}{\sin \left( A + B \right)} = \frac{x - 1}{x + 1}\]
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).
Prove that:
If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.
Find the maximum and minimum values of each of the following trigonometrical expression:
12 cos x + 5 sin x + 4
Reduce each of the following expressions to the sine and cosine of a single expression:
cos x − sin x
Show that sin 100° − sin 10° is positive.
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.
If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).
tan 20° + tan 40° + \[\sqrt{3}\] tan 20° tan 40° is equal to
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 tan 69° + tan 66° − tan 69° tan 66° = 2k, then k =
If \[\tan\alpha = \frac{x}{x + 1}\] and \[\tan\alpha = \frac{x}{x + 1}\], then \[\alpha + \beta\] is equal to
If 3 tan (θ – 15°) = tan (θ + 15°), 0° < θ < 90°, then θ = ______.
Find the most general value of θ satisfying the equation tan θ = –1 and cos θ = `1/sqrt(2)`.
If tan θ = 3 and θ lies in third quadrant, then the value of sin θ ______.
The value of tan3A - tan2A - tanA is equal to ______.
If tanθ = `a/b`, then bcos2θ + asin2θ is equal to ______.
3(sinx – cosx)4 + 6(sinx + cosx)2 + 4(sin6x + cos6x) = ______.
