Question

find the value of x,
$\rm 8^{1+\sin x + \sin ^2 x + \sin ^3 x +... ∞} = 64$
​

Answer 1

Given :-

→ 8^(1+sin x+ sin² x+sin³ x +...∞) = 64

To find :-

→ The value of x

Solution :-

Given that

8^(1+sin x+ sin² x+sin³ x +...∞) = 64

On taking 1+sin x + sin² x + sin³ x + ...∞

First term (a) = 1

Common ratio (r) = sin x/1 = sin x

r = sin² x / sin x = sin x

Since, the common ratio is same throughout the series.So the series in the Geometric Progression.

We know that

The sum of infinite terms in the G.P

S ∞ = a/(1-r)

Therefore,

S ∞ = 1/(1-sin x)

Now,

Given equation becomes

8^ [1/(1-sin x) ] = 64

=> 8^ [1/(1-sin x) ] = 8²

If the bases are equal then exponents must be equal.

=> 1/(1- sin x) = 2

=> 2 (1- sin x ) = 1

=> 1-sin x = 1/2

=> sin x = 1-(1/2)

=> sin x = (2-1)/2

=> sin x = 1/2

=> sin x = sin 30°

=> x = 30° or

=> x = 30×(π/180)©

=> x = π/6©

Therefore, X = 30° or π/6©

Answer :-

The value of x is 30° or π/6©

Used formulae:-

→ The sum of infinite terms in an GP is

S∞ = a/(1-r)

• a = First term
• r = Common ratio

→ If a^m = a^n => m = n

→ π© = 180°

• © = radians

Answer 2

Step-by-step explanation:

Given :-

→ 8^(1+sin x+ sin² x+sin³ x +...∞) = 64

To find :-

→ The value of x

Solution :-

Given that

8^(1+sin x+ sin² x+sin³ x +...∞) = 64

On taking 1+sin x + sin² x + sin³ x + ...∞

First term (a) = 1

Common ratio (r) = sin x/1 = sin x

r = sin² x / sin x = sin x

Since, the common ratio is same throughout the series.So the series in the Geometric Progression.

We know that

The sum of infinite terms in the G.P

S ∞ = a/(1-r)

Therefore,

S ∞ = 1/(1-sin x)

Now,

Given equation becomes

8^ [1/(1-sin x) ] = 64

=> 8^ [1/(1-sin x) ] = 8²

If the bases are equal then exponents must be equal.

=> 1/(1- sin x) = 2

=> 2 (1- sin x ) = 1

=> 1-sin x = 1/2

=> sin x = 1-(1/2)

=> sin x = (2-1)/2

=> sin x = 1/2

=> sin x = sin 30°

=> x = 30° or

=> x = 30×(π/180)©

=> x = π/6©

Therefore, X = 30° or π/6©

Answer :-

The value of x is 30° or π/6©

Used formulae:-

→ The sum of infinite terms in an GP is

S∞ = a/(1-r)

a = First term

r = Common ratio

→ If a^m = a^n => m = n

→ π© = 180°

© = radians

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