Question

1. If 1 / (cot 2A) = cot(A - 18 degrees) , where A is an acute angle, find the value of A.​

Answer 1

Answer:

Summary: If tan 2A = cot (A - 18°), where 2A is an acute angle, then the value of A is 36°.

Answer 2

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

Given that,

$\rm :\longmapsto\:\dfrac{1}{cot2A} = cot(A - 18\degree)$

and

$\rm :\longmapsto\:0\degree < A < 90\degree$

We know,

$\purple{\rm :\longmapsto\:\boxed{\tt{ \frac{1}{cotx} = tanx}}}$

So, using this identity, we get

$\rm :\longmapsto\:tan2A = cot(A - 18\degree)$

We know that,

$\purple{\rm :\longmapsto\:\boxed{\tt{ tan(90\degree - x) = cotx}}}$

So, using this identity, we get

$\rm :\longmapsto\:cot(90\degree - 2A) = cot(A - 18\degree)$

So, on comparing, we get

$\rm :\longmapsto\:90\degree - 2A = A - 18\degree$

$\rm :\longmapsto\:2A + A = 90\degree +18\degree$

$\rm :\longmapsto\:3A = 108\degree$

$\bf\implies \:A \: = \: 36\degree$

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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