Question

Q1.3tanø=secø find cotø​.

Answer 1

Step-by-step explanation:

GIVEN :-

3tan∅ = sec∅.

TO FIND :-

The value of cot∅.

SOLUTION :-

Now we have to convert all the terms in the form of sin∅ and cos∅.

So as we know the following trigonometric identities :

★ sec∅ = 1/cos∅ or cos∅ = 1/sec∅ ★

★ cot∅ = cos∅/sin∅ or cot∅ = 1/tan∅ ★

★ tan∅ = sin∅/cos∅ ★

★ cos∅ = √(1 - sin²∅) ★

Now substitute the trigonometric identities in the given equation,

$\begin{gathered} \implies \displaystyle \sf \: 3 \tan( \theta) = \sec( \theta) \\ \end{gathered}$

$\begin{gathered} \implies \displaystyle \sf \: 3 \times \frac{ \sin( \theta) }{ \cos( \theta) } = \frac{1}{ \cos( \theta) } \\ \end{gathered}$

On cancelling cos∅ with cos∅ we get,

$\begin{gathered} \\ \implies \displaystyle \sf \:3 \sin( \theta) = 1 \\ \end{gathered}$

$\begin{gathered}\implies \displaystyle \sf \: \sin( \theta) = \frac{1}{3} \: \dashrightarrow(1) \\ \end{gathered}$

Now as we know the identity :- cos∅ = √(1 - sin²∅).

$\begin{gathered}\implies \displaystyle \sf \: \cot( \theta) = \frac{ \cos( \theta) }{ \sin( \theta) } \\ \end{gathered}$

$\begin{gathered}\implies \displaystyle \sf \: \cot( \theta) = \frac{ \sqrt{1 - \sin ^{2} ( \theta) } }{ \bigg(\dfrac{1}{3} \bigg) } \\ \end{gathered}$

$\begin{gathered}\implies \displaystyle \sf \: \cot( \theta) = \frac{ \sqrt{1 - \bigg( \dfrac{1}{3} \bigg) ^{2} } }{ \bigg( \dfrac{1}{3} \bigg)} \\ \end{gathered}$

$\begin{gathered}\implies \displaystyle \sf \: \cot( \theta) = \frac{ \sqrt{1 - \dfrac{1}{9} } }{ \bigg( \dfrac{1}{3} \bigg)} \\ \end{gathered}$

$\begin{gathered}\implies \displaystyle \sf \: \cot( \theta) = \frac{ \sqrt{ \dfrac{8}{9} } }{ \bigg( \dfrac{1}{3} \bigg)} \\ \end{gathered}$

$\begin{gathered}\implies \displaystyle \sf \: \cot( \theta) = \frac{ \dfrac{2}{3} \sqrt{ 2 } }{ \bigg( \dfrac{1}{3} \bigg)} \\ \end{gathered}$

$\implies \displaystyle \underline{ \boxed{ \sf \: \cot( \theta) =2 \sqrt{2} }}$