Advertisements
Advertisements
Question
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
Solution
\[\text{ Given }: \]
\[x \cos\theta = y\left( \cos\theta\cos\frac{2\pi}{3} - \sin\theta \sin\frac{2\pi}{3} \right) = z\left( \cos\theta\cos\frac{4\pi}{3} - \sin\theta \sin\frac{4\pi}{3} \right)\]
\[ \Rightarrow x\cos\theta = y\left( - \frac{1}{2}\cos\theta - \frac{\sqrt{3}}{2}\sin\theta \right) = z\left( - \frac{1}{2}\cos\theta + \frac{\sqrt{3}}{2}\sin\theta \right) \]
\[ \Rightarrow x = \frac{y}{2}\left( - 1 - \sqrt{3}\tan\theta \right) = \frac{z}{2}\left( - 1 + \sqrt{3}\tan\theta \right)\]
\[x = \frac{y}{2}\left( - 1 - \sqrt{3}\tan\theta \right)\]
\[z = \frac{y\left( - 1 - \sqrt{3}\tan\theta \right)}{\left( - 1 + \sqrt{3}\tan\theta \right)}\]
\[\text{ Now }, \]
\[\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{2}{y\left( - 1 - \sqrt{3}\tan\theta \right)} + \frac{1}{y} + \frac{\left( - 1 + \sqrt{3}\tan\theta \right)}{y\left( - 1 - \sqrt{3}\tan\theta \right)}\]
\[ = \frac{2 + \left( - 1 - \sqrt{3}\tan\theta \right) + \left( - 1 + \sqrt{3}\tan\theta \right)}{y\left( - 1 - \sqrt{3}\tan\theta \right)}\]
\[ = 0\]
APPEARS IN
RELATED QUESTIONS
Prove that: `2 sin^2 (3pi)/4 + 2 cos^2 pi/4 + 2 sec^2 pi/3 = 10`
Prove the following:
`cos ((3pi)/ 2 + x ) cos(2pi + x) [cot ((3pi)/2 - x) + cot (2pi + x)]= 1`
Prove that: `(cos x + cos y)^2 + (sin x - sin y )^2 = 4 cos^2 (x + y)/2`
Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
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:
sin (A − B)
If \[\sin A = \frac{3}{5}, \cos B = - \frac{12}{13}\], where A and B both lie in second quadrant, find the value of 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)
If \[\tan A = \frac{3}{4}, \cos B = \frac{9}{41}\], where π < A < \[\frac{3\pi}{2}\] and 0 < B <\[\frac{\pi}{2}\], find tan (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°
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)
Prove that:
\[\frac{7\pi}{12} + \cos\frac{\pi}{12} = \sin\frac{5\pi}{12} - \sin\frac{\pi}{12}\]
Prove that
If \[\tan A = \frac{5}{6}\text{ and }\tan B = \frac{1}{11}\], prove that \[A + B = \frac{\pi}{4}\].
Prove that:
\[\frac{\sin \left( A - B \right)}{\cos A \cos B} + \frac{\sin \left( B - C \right)}{\cos B \cos C} + \frac{\sin \left( C - A \right)}{\cos C \cos A} = 0\]
Prove that:
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
Prove that:
\[\frac{\tan^2 2x - \tan^2 x}{1 - \tan^2 2x \tan^2 x} = \tan 3x \tan x\]
Prove that sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin A.
If cos A + sin B = m and sin A + cos B = n, prove that 2 sin (A + B) = m2 + n2 − 2.
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\[\sin\theta = \frac{\lambda + 1}{\lambda - 1}\sin\phi\]
Find the maximum and minimum values of each of the following trigonometrical expression:
12 sin x − 5 cos x
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
If a = b \[\cos \frac{2\pi}{3} = c \cos\frac{4\pi}{3}\] then write the value of ab + bc + ca.
The maximum value of \[\sin^2 \left( \frac{2\pi}{3} + x \right) + \sin^2 \left( \frac{2\pi}{3} - x \right)\] is
If cos (A − B) \[= \frac{3}{5}\] and tan A tan B = 2, then
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))]
If tan θ = 3 and θ lies in third quadrant, then the value of sin θ ______.
If tanα = `m/(m + 1)`, tanβ = `1/(2m + 1)`, then α + β is equal to ______.
The value of sin(45° + θ) - cos(45° - θ) is ______.
The value of `cot(pi/4 + theta)cot(pi/4 - theta)` is ______.
If tanθ = `a/b`, then bcos2θ + asin2θ is equal to ______.
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 tan(π cosθ) = cot(π sinθ), then `cos(theta - pi/4) = +- 1/(2sqrt(2))`.
