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$ \Large \mathtt \red{Question :}$
Find the general solution for $ \rm{cot^{2}\theta + \frac{3}{sin\theta}+3 = 0}$.

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Answer 1

• $\sf\pink{Given\::cot^{2}\theta\:+\:\frac{3}{\sin\theta}\:+\:3\:=\:0}$

• $\sf\blue{To\:Find\::General\: Solution\:of\:this\: equation}$

$\sf\red{Solution\::}$

• $\sf{\cot^{2}\theta\:+\:\frac{3}{\sin\theta}\:+\:3\:=\:0}$

We know that cot = cos/sin so put that value over here which will give you the following equation ,

• $\sf{\frac{\cos^2\theta}{\sin^2\theta}\:+\:\frac{3}{\sin\theta}\:+\:3\:=\:0}$

Now by using the first identity of trigonometry we get here ,

• $\sf{\cos^2\theta\:+\:\sin^2\theta\:=\:1}$

Or

• $\sf{\cos^2\theta\:=\:1-\:\sin^2\theta\:}$

Now use this value here ,

• $\sf{\frac{1\:-\:\sin^2\theta}{\sin^2\theta}\:+\:\frac{3}{\sin\theta}\:+\:3\:=\:0}$

Now multiply LHS with sin²theta

• $\sf{1\:-\:\sin^2\theta\:+\:3\sin\theta\:+\:3\sin^2\theta\:=\:0}$

• $\sf{1\:+\:+\:3\sin\theta\:+\:2\sin^2\theta\:=\:0}$

We can write it as ,

• $\sf{2\sin^2\theta\:+\:2\sin\theta\:+\:\sin\theta\:+\:1\:=\:0}$

• = (2 sin theta + 1 )( sin theta + 1 ) = 0

• = sin theta = - 1/2 or - 1

• $\sf{\theta\:=\:n\pi\:+\:(-1)^n\:\frac{-\pi}{6}\:and\:\theta\:=\:n\pi\:+\:(-1)^n\:\frac{-\pi}{2}}$

Answer 2

Given :-

• Cot²A + 3/SinA + 3 = 0

To Find :-

• General Solution for A = ?

Solution :-

• To calculate the solution at first we have to simplify by simplifying expression by applying formula.

Calculation begins :-

⇒ Cot²A + 3/SinA + 3 = 0

⇒ Cos²A/Sin²A + 3/SinA + 3/1 = 0

⇒ Cos²A + 3SinA + 3Sin²A/Sin²A = 0

⇒ Cos² A + 3SinA + 3Sin²A = 0

⇒ Cos²A + 3SinA + Sin²A + 2Sin²A = 0

⇒ Cos²A + Sin²A + 3SinA + 2Sin²A = 0

• Cos²A + Sin²A = 1

⇒ 2Sin²A + 3SinA + 1 = 0

• Splitting the middle term here :-

⇒ 2Sin²A + 2SinA + SinA + 1 = 0

⇒ 2SinA(SinA + 1) + 1(SinA + 1) = 0

⇒ (2SinA + 1)(SinA + 1) = 0

• Now calculate general Solution for A :-

⇒ 2SinA + 1 = 0 ⇒ SinA = -1/2

⇒ SinA = -(π/6) ⇒ A = nπ + (-1)^n (-π/6)

⇒ SinA + 1 = 0 ⇒ SinA = -1

⇒ SinA = -(π/2) ⇒ A = nπ + (-1)^n (-π/2)

Hence,

• General Solution for this expression :-

⇒ A = nπ + (-1)^n (-π/6) or nπ + (-1)^n (-π/2)