Question

3tanø=secø find cotø​

Answer 1

GIVEN :-

• 3tan∅ = sec∅.

TO FIND :-1

• 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,

$\implies \displaystyle \sf \: 3 \tan( \theta) = \sec( \theta)$

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

On cancelling cos∅ with cos∅ we get,

$\\ \implies \displaystyle \sf \:3 \sin( \theta) = 1$

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

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

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

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

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

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

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

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

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

Answer 2

Given ;-

• 3 tanø = secø

To Find :-

• cotø

Formula Used ;-

$\longmapsto \boxed{ \blue{ \bf \: tanx \: = \: \dfrac{sinx}{cosx} }}$

$\longmapsto \boxed{ \blue{ \bf \: secx \: = \: \dfrac{1}{cosx} }}$

$\longmapsto \boxed{ \blue{ \bf \: {cosec}^{2} x - {cot}^{2} x = 1}}$

$\large\underline\purple{\bold{Solution :- }}$

Given that

$\rm :\implies\:3 \: tan \phi \: = sec\phi \:$

$\rm :\implies\:3 \: \dfrac{sin\phi \: }{ \cancel{cos\phi \:} } = \dfrac{1}{ \cancel{cos\phi \:} }$

$\rm :\implies\:sin\phi \: = \dfrac{1}{3}$

$\rm :\implies \: \boxed{ \green {\bf\:cosec\phi \: = 3} }- - (1)$

Now,

We know that,

$\rm :\implies\: {cosec}^{2} \phi \: - {cot}^{2} \phi \: = 1$

$\rm :\implies\: {(3)}^{2} - {cot}^{2} \phi \: = 1$

$\rm :\implies\: {cot}^{2} \phi \: = 9 - 1$

$\rm :\implies\: {cot}^{2} \phi \: = 8$

$\rm :\implies\:cot\phi \: = \sqrt{8}$

$\rm :\implies \: \boxed{ \green{ \bf\:cot\phi \: = 2 \sqrt{2} }}$

Additional Information:-

Additional Information:- Relationship between sides and T ratios

• sin θ = Opposite Side/Hypotenuse
• cos θ = Adjacent Side/Hypotenuse
• tan θ = Opposite Side/Adjacent Side
• sec θ = Hypotenuse/Adjacent Side
• cosec θ = Hypotenuse/Opposite Side
• cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ
• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ

Co-function Identities

• sin (90°−x) = cos x
• cos (90°−x) = sin x
• tan (90°−x) = cot x
• cot (90°−x) = tan x
• sec (90°−x) = cosec x
• cosec (90°−x) = sec x

Fundamental Trigonometric Identities

• sin²θ + cos²θ = 1
• sec²θ - tan²θ = 1
• cosec²θ - cot²θ = 1