Advertisements
Advertisements
प्रश्न
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`]
Advertisements
उत्तर
Given that: `(sqrt(3) - 1) costheta + (sqrt(3) + 1) sin theta` = 2
Put `sqrt(3) - 1` = r sinα, `sqrt(3) + 1` = r cosα
By squaring and adding, we get
r2 = `3 + 1 - 2sqrt(3) + 3 + 1 + 2sqrt(3)`
⇒ r2 = 8
⇒ r = `+- 2sqrt(2)`
Now the given equation can be written as
rsinα cosθ + rcosα sinθ = 2
⇒ r(sinα cosθ + cosα sinθ) = 2
⇒ `2sqrt(2) sin(alpha + theta)` = 2
⇒ `sin(alpha + theta) = 2/(2sqrt(2)) = 1/sqrt(2)`
⇒ `sin(alpha + theta) = sin pi/4`
∴ α + θ = `npi + (-1)^n * pi/4` .....(i)
Now `(r sin alpha)/(r cos alpha) = (sqrt(3) - 1)/(sqrt(3) + 1)`
⇒ tanα = `(tan pi/3 - tan pi/4)/(1 + tan pi/4 * tan pi/3)`
⇒ tanα = `tan(pi/3 - pi/4)`
⇒ tanα = `tan pi/12`
∴ α = `pi/12`
Putting the value of α in equation (i) we get
`pi/12 + theta = npi + (-1)^n * pi/4`
∴ θ = `npi + (-1)^n * pi/4 - pi/12`
Hence, the general solution of the given equation is θ = `npi + (-1)^n * pi/4 - pi/12`, n ∈ Z.
APPEARS IN
संबंधित प्रश्न
Prove that `2 sin^2 pi/6 + cosec^2 (7pi)/6 cos^2 pi/3 = 3/2`
Prove the following: `cos (pi/4 xx x) cos (pi/4 - y) - sin (pi/4 - x)sin (pi/4 - y) = sin (x + y)`
Prove the following:
`(cos (pi + x) cos (-x))/(sin(pi - x) cos (pi/2 + x)) = cot^2 x`
Prove the following:
sin2 6x – sin2 4x = sin 2x sin 10x
Prove the following:
`(cos9x - cos5x)/(sin17x - sin 3x) = - (sin2x)/(cos 10x)`
Prove the following:
`(cos 4x + cos 3x + cos 2x)/(sin 4x + sin 3x + sin 2x) = cot 3x`
Prove the following:
cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1
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:
cos (A + B)
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)
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:
cos 47° cos 13° − sin 47° sin 13°
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)
If \[\tan A = \frac{5}{6}\text{ and }\tan B = \frac{1}{11}\], prove that \[A + B = \frac{\pi}{4}\].
Prove that: \[\frac{\sin \left( A + B \right) + \sin \left( A - B \right)}{\cos \left( A + B \right) + \cos \left( A - B \right)} = \tan A\]
Prove that:
sin2 B = sin2 A + sin2 (A − B) − 2 sin A cos B sin (A − B)
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 sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin A.
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).
Prove that:
\[\frac{1}{\sin \left( x - a \right) \sin \left( x - b \right)} = \frac{\cot \left( x - a \right) - \cot \left( x - b \right)}{\sin \left( a - b \right)}\]
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 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 \[\frac{\cos \left( x - y \right)}{\cos \left( x + y \right)} = \frac{m}{n}\] then write the value of tan x tan y.
If a = b \[\cos \frac{2\pi}{3} = c \cos\frac{4\pi}{3}\] then write the value of ab + bc + ca.
The value of \[\sin^2 \frac{5\pi}{12} - \sin^2 \frac{\pi}{12}\]
If in ∆ABC, tan A + tan B + tan C = 6, then cot A cot B cot C =
tan 3A − tan 2A − tan A =
If \[\cos P = \frac{1}{7}\text{ and }\cos Q = \frac{13}{14}\], where P and Q both are acute angles. Then, the value of P − Q is
If tan (A − B) = 1 and sec (A + B) = \[\frac{2}{\sqrt{3}}\], the smallest positive value of B is
Express the following as the sum or difference of sines and cosines:
2 cos 3x sin 2xa
If sinθ + cosθ = 1, then find the general value of θ.
If cotθ + tanθ = 2cosecθ, then find the general value of θ.
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 sinθ + cosecθ = 2, then sin2θ + cosec2θ is equal to ______.
The value of tan 75° - cot 75° is equal to ______.
If sinx + cosx = a, then sin6x + cos6x = ______.
