Advertisements
Advertisements
Question
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`]
Solution
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
RELATED QUESTIONS
Prove that: `sin^2 pi/6 + cos^2 pi/3 - tan^2 pi/4 = -1/2`
Prove that `2 sin^2 pi/6 + cosec^2 (7pi)/6 cos^2 pi/3 = 3/2`
Prove that `cot^2 pi/6 + cosec (5pi)/6 + 3 tan^2 pi/6 = 6`
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:
`(sin 5x + sin 3x)/(cos 5x + cos 3x) = tan 4x`
Prove the following:
`(sin x - siny)/(cos x + cos y)= tan (x -y)/2`
Prove the following:
cos 6x = 32 cos6 x – 48 cos4 x + 18 cos2 x – 1
Prove that: `(cos x - cosy)^2 + (sin x - sin y)^2 = 4 sin^2 (x - y)/2`
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)
Prove that \[\frac{\tan 69^\circ + \tan 66^\circ}{1 - \tan 69^\circ \tan 66^\circ} = - 1\].
Prove that:
cos2 A + cos2 B − 2 cos A cos B cos (A + B) = sin2 (A + B)
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
If 12 sin x − 9sin2 x attains its maximum value at x = α, then write the value of sin α.
If A + B = C, then write the value of tan A tan B tan C.
tan 20° + tan 40° + \[\sqrt{3}\] tan 20° tan 40° is equal to
The value of \[\cos^2 \left( \frac{\pi}{6} + x \right) - \sin^2 \left( \frac{\pi}{6} - x \right)\] is
If A − B = π/4, then (1 + tan A) (1 − tan B) is equal to
If tan 69° + tan 66° − tan 69° tan 66° = 2k, then k =
Express the following as the sum or difference of sines and cosines:
2 sin 3x cos x
Express the following as the sum or difference of sines and cosines:
2 cos 3x sin 2xa
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)`.
Match each item given under column C1 to its correct answer given under column C2.
| C1 | C2 |
| (a) `(1 - cosx)/sinx` | (i) `cot^2 x/2` |
| (b) `(1 + cosx)/(1 - cosx)` | (ii) `cot x/2` |
| (c) `(1 + cosx)/sinx` | (iii) `|cos x + sin x|` |
| (d) `sqrt(1 + sin 2x)` | (iv) `tan x/2` |
If cos(θ + Φ) = m cos(θ – Φ), then prove that 1 tan θ = `(1 - m)/(1 + m) cot phi`
[Hint: Express `(cos(theta + Φ))/(cos(theta - Φ)) = m/1` and apply Componendo and Dividendo]
If tan θ = 3 and θ lies in third quadrant, then the value of sin θ ______.
The value of tan 75° - cot 75° is equal to ______.
If α + β = `pi/4`, then the value of (1 + tan α)(1 + tan β) is ______.
If sinx + cosx = a, then sin6x + cos6x = ______.
State whether the statement is True or False? Also give justification.
If cosecx = 1 + cotx then x = 2nπ, 2nπ + `pi/2`
