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General Solut...

General Solution of tan theta = tan alpha

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Q.

Which of the following statements are correct?

1 . if sin θ = sin α ⇒ θ = nπ + $(- 1)^{n}$ α where α ∈ [ - $\frac{π}{2}$ , $\frac{π}{2}$ ] n ∈ I

2 . if cos θ = cos α ⇒ 2nπ±α where α ∈ [0, π] n ∈ I

3 . if tan θ = tan α ⇒ θ = nπ + α where α ∈ (- $\frac{π}{2}$ , $\frac{π}{2}$ ) n ∈ I

Only 1 & 2
Only 1 & 3
Only 2 & 3
All 1, 2 & 3

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Q.

The number of real roots of the equation tan (x + \frac{π}{6}) = 2 tan x, for x \in (0, 3 π) are

.

\N
1
2

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Q.

If tan $α = 2$ , then the values of x which satisfy the relation

$t a n x = \frac{1}{2} a r e$ $0 < x <$ $2 π a n d$ $0 < α <$ $\frac{π}{2}$

$\frac{π}{2} + α$
$π + α$
$\frac{3 π}{2} - α$
$\frac{π}{2} - α$

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Q.

$\begin{matrix} T r i g o n o m e t r i c E q u a t i o n s & G e n e r a l S o l u t i o n s 1. t a n x = 2 & P . 2 n π \pm \frac{2 π}{3}, n \in I 2. s i n x = \frac{\sqrt{3}}{2} & Q . n π - \frac{π}{4}, n \in I 3. c o s x = \frac{- 1}{2} & R . n π + (- 1)^{n} \frac{π}{3}, n \in I 4. c o t x = - 1 & S . n π + t a n^{- 1} (2), n \in I \end{matrix}$

1 - R, 2 - S, 3 - Q, 4 - P
1 - R, 2 - S, 3 - P, 4 - Q
1 - S, 2 - R, 3 - Q, 4 - P
1 - S, 2 - R, 3 - P, 4 - Q

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Q. The general solution of the equation

tan 5 θ = cot 3 θ

is given by

$θ = \frac{n π}{8} - \frac{π}{16}, n \in Z .$
$θ = \frac{n π}{8} + \frac{π}{16}, n \in Z .$
$θ = \frac{n π}{8} + \frac{π}{8}, n \in Z .$
$θ = \frac{n π}{8} - \frac{π}{8}, n \in Z .$

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Q.

If α is the value of $x ϵ [0, π]$ satisfying 3 cos x + 3 sin x + sin 3x - cos 3x = 0, then find the value of $\frac{4 α}{π}$ ?

___

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Q. The general solution of the equation

tan 5 θ = cot 3 θ

is given by

$θ = \frac{n π}{8} - \frac{π}{16}, n \in Z .$
$θ = \frac{n π}{8} + \frac{π}{16}, n \in Z .$
$θ = \frac{n π}{8} + \frac{π}{8}, n \in Z .$
$θ = \frac{n π}{8} - \frac{π}{8}, n \in Z .$

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Q. The complete solution set of

sin x - \sqrt{3} cos x = 0

is

$n π - \frac{π}{3}$
$n π \pm \frac{π}{3}$
$n π + \frac{π}{3}$
$n π \pm \frac{π}{6}$

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Q.

Which of the following statements are correct?

1 . if sin θ = sin α ⇒ θ = nπ + $(- 1)^{n}$ α where α ∈ [ - $\frac{π}{2}$ , $\frac{π}{2}$ ] n ∈ I

2 . if cos θ = cos α ⇒ 2nπ±α where α ∈ [0, π] n ∈ I

3 . if tan θ = tan α ⇒ θ = nπ + α where α ∈ (- $\frac{π}{2}$ , $\frac{π}{2}$ ) n ∈ I

Only 1 & 2
Only 1 & 3
Only 2 & 3
All 1, 2 & 3

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Q.

If the general solution of the equation tan 3 θ = - 1, is given by θ = \frac{n π}{3} - α, where n \in Z and α \in (0, \frac{π}{6}], then α equals

.

$\frac{π}{6}$
$\frac{π}{12}$
$\frac{π}{15}$

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Q. If

cot θ - p q tan θ = p - q; p, q \neq 0

, then the general value of

θ

is

$m π + {tan}^{- 1} (\frac{1}{q}), m \in Z$
$m π + {tan}^{- 1} (\frac{- 1}{p}), m \in Z$
$m π + {tan}^{- 1} (\frac{1}{p q}), m \in Z$
$m π + {tan}^{- 1} (\frac{1}{p}), m \in Z$

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Q. The number of solutions of the equation

| 5 tan 2 θ tan θ - 12 tan θ | + ∣ ∣ 3 {sin}^{2} θ - sin 2 θ ∣ ∣ = 0

in the interval

[0, 2 π]

is

$2$
$3$
$5$
$6$

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Q. The general value of

θ

satisfying the equation

t a n^{2} θ + s e c 2 θ = 1

is___

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Q.

If tan (\frac{p π}{4}) = c o t (\frac{qπ}{4}), then the value of (p + q) is

.

$2 (2 n + 1), n \in Z .$
$(2 n - 1), n \in Z .$
$2 n + \frac{1}{2}, n \in Z .$
$n + \frac{1}{2}, n \in Z .$

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Q. The general solution of the equation

tan θ = tan α, \forall α \in (- \frac{π}{2}, \frac{π}{2})

is given by

θ = 2 n π + α, n \in Z .

True
False

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Q.

How many values of $x ϵ [0, 2 π]$ satisfies the equation sin 2x + 5 sin x + 1 + 5 cos x = 0?

___

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Q. The general solution of the equation sin x -3 sin 2x + sin 3x =cos x -3 cos 2x + cos 3x is

(n \in Z)

$n π + \frac{π}{4}$
$\frac{n π}{2} + \frac{π}{8}$
$(- 1)^{n} \frac{n π}{2} + \frac{π}{8}$
$2 n π + c o s^{- 1} \frac{2}{3}$

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