Question

if 1+cot²(3A-20)=cosec² 52°, where 3A is an acute angle then what is the value of A​

Answer 1

$\large\underline{\sf{Solution-}}$

Given Trigonometric equation is

$\rm \: 1 + {cot}^{2}(3A - 20 \degree) \: = \: {cosec}^{2}52\degree$

Also, given that

$\rm \: 3A \: is \: an \: acute \: angle$

$\rm\implies \:\rm \: 3A - 20\degree \: is \: also\: an \: acute \: angle$

$\rm\implies \:cot(3A - 20\degree ) > 0$

Now, Consider the given Trigonometric equation

$\rm \: 1 + {cot}^{2}(3A - 20 \degree) \: = \: {cosec}^{2}52\degree$

can be rewritten as

$\rm \:{cot}^{2}(3A - 20 \degree) \: = \: {cosec}^{2}52\degree - 1$

We know that,

$\boxed{\sf{ \: {cosec}^{2}x - {cot}^{2}x = 1 \: \: }}$

So, using this, we get

$\rm \: {cot}^{2}(3A - 20\degree ) = {cot}^{2}52\degree$

$\rm\implies \:cot(3A - 20\degree ) = cot52\degree$

So, on comparing, we get

$\rm \: 3A - 20\degree = 52\degree$

$\rm \: 3A = 52\degree + 20\degree$

$\rm \: 3A = 72\degree$

$\rm\implies \:A = 24\degree$

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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

Answer 2

${ \huge{ \boxed{ \bf{\underline{ \red{Answer}}}}}} : -$

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