Advertisements
Advertisements
Question
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`
Options
True
False
Advertisements
Solution
This statement is True.
Explanation:
Given that: tanθ + tan2θ + `sqrt(3)` tanθ tan2θ = `sqrt(3)`
⇒ tanθ + tan2θ = `sqrt(3) - sqrt(3) tan theta tan 2theta`
⇒ tanθ + tan2θ = `sqrt(3) (1 - tan theta tan 2theta)`
⇒ `(tan theta + tan 2theta)/(1 - tan theta tan 2theta) = sqrt(3)`
⇒ tan(θ + 2θ) = `sqrt(3)`
⇒ tan3θ = `tan pi/3`
∴ 3θ = `"n"pi + pi/3`
So θ = `("n"pi)/3 + pi/9`
APPEARS IN
RELATED QUESTIONS
Prove the following:
`cos ((3pi)/4 + x) - cos((3pi)/4 - x) = -sqrt2 sin x`
Prove the following:
cos2 2x – cos2 6x = sin 4x sin 8x
Prove the following:
sin 2x + 2sin 4x + sin 6x = 4cos2 x sin 4x
Prove the following:
cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x)
Prove the following:
`(cos9x - cos5x)/(sin17x - sin 3x) = - (sin2x)/(cos 10x)`
Prove the following:
`tan 4x = (4tan x(1 - tan^2 x))/(1 - 6tan^2 x + tan^4 x)`
Prove that: `(cos x + cos y)^2 + (sin x - sin y )^2 = 4 cos^2 (x + y)/2`
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 \[\cos A = - \frac{24}{25}\text{ and }\cos B = \frac{3}{5}\], where π < A < \[\frac{3\pi}{2}\text{ and }\frac{3\pi}{2}\]< B < 2π, find the following:
sin (A + B)
Prove that
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 α, β are two different values of x lying between 0 and 2π, which satisfy the equation 6 cos x + 8 sin x = 9, find the value of sin (α + 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
Find the maximum and minimum values of each of the following trigonometrical expression:
\[5 \cos x + 3 \sin \left( \frac{\pi}{6} - x \right) + 4\]
Find the maximum and minimum values of each of the following trigonometrical expression:
sin x − cos x + 1
Write the maximum value of 12 sin x − 9 sin2 x.
If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).
If \[\tan A = \frac{a}{a + 1}\text{ and } \tan B = \frac{1}{2a + 1}\]
If in ∆ABC, tan A + tan B + tan C = 6, then cot A cot B cot C =
If cot (α + β) = 0, sin (α + 2β) is equal to
The value of \[\cos^2 \left( \frac{\pi}{6} + x \right) - \sin^2 \left( \frac{\pi}{6} - x \right)\] is
If tan θ1 tan θ2 = k, then \[\frac{\cos \left( \theta_1 - \theta_2 \right)}{\cos \left( \theta_1 + \theta_2 \right)} =\]
If sin (π cos x) = cos (π sin x), then sin 2x = ______.
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 =
Show that 2 sin2β + 4 cos (α + β) sin α sin β + cos 2(α + β) = cos 2α
If angle θ is divided into two parts such that the tangent of one part is k times the tangent of other, and Φ is their difference, then show that sin θ = `(k + 1)/(k - 1)` sin Φ
If cotθ + tanθ = 2cosecθ, then find the general value of θ.
If sinθ + cosecθ = 2, then sin2θ + cosec2θ is equal to ______.
If tanA = `1/2`, tanB = `1/3`, then tan(2A + B) is equal to ______.
If sinx + cosx = a, then |sinx – cosx| = ______.
Given x > 0, the values of f(x) = `-3cos sqrt(3 + x + x^2)` lie in the interval ______.
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))`.
