If-0-x-pi-and-81-sin-2-x-81-cos-2-x-30-then-x-is-equal-to-

Question Number 59389 by hovea cw last updated on 09/May/19

If 0 ⩽ x ⩽ π and 81^{\sin^{2} x} + 81^{\cos^{2} x} = 30,

then x is equal to

Answered by ajfour last updated on 09/May/19

t + \frac{81}{t} = 30

\Rightarrow t^{2} - 30 t + 81 = 0

\Rightarrow t = 3, 27

81^{\sin^{2} x} = 81^{\frac{1}{4}}, 81^{\frac{3}{4}}

\Rightarrow x = \frac{π}{6}, \frac{π}{3}, \frac{2 π}{3}, \frac{5 π}{6} .

Answered by tanmay last updated on 09/May/19

{(81)}^{{s i n}^{2} x} + {(81)}^{1 - {s i n}^{2} x} = 30

{(81)}^{{s i n}^{2} x} + \frac{81}{{(81)}^{{s i n}^{2} x}} = 30

{(9)}^{2 {s i n}^{2} x} + \frac{9^{2}}{{(9)}^{2 {s i n}^{2} x}} = 30

{(9^{{s i n}^{2} x} - \frac{9}{9^{{s i n}^{2} x}})}^{2} + 2 \times 9^{{s i n}^{2} x} \times \frac{9}{9^{{s i n}^{2} x}} = 30

{(9^{{s i n}^{2} x} - \frac{9}{9^{{s i n}^{2} x}})}^{2} = {(2 \sqrt{3})}^{2}

{(k - \frac{9}{k})}^{2} - {(2 \sqrt{3})}^{2} = 0

k - \frac{9}{k} + 2 \sqrt{3} = 0

k^{2} + 2 \sqrt{3} k - 9 = 0

k^{2} + (3 \sqrt{3} - \sqrt{3}) k - 3 \times \sqrt{3} \times \sqrt{3} = 0

k (k + 3 \sqrt{3}) - \sqrt{3} (k + 3 \sqrt{3}) = 0

(k + 3 \sqrt{3}) (k - \sqrt{3}) = 0

k + 3 \sqrt{3} \neq 0

k = \sqrt{3}

9^{{s i n}^{2} x} = 3^{\frac{1}{2}}

3^{2 {s i n}^{2} x} = 3^{\frac{1}{2}}

{s i n}^{2} x = \frac{1}{4}

s i n x = \frac{1}{2} = s i n (\frac{π}{6}) \to x = \frac{π}{6} (f i r s t q u a d r a n t)

a n d (π - \frac{π}{6}) i, e \frac{5 π}{6} i n s e c o n d q u a d r a n t

s i n x \neq - \frac{1}{2} w h e n π ⩾ x ⩾ 0

a g a i n

k - \frac{9}{k} - 2 \sqrt{3} = 0

k^{2} - 9 - 2 \sqrt{3} k = 0

k^{2} - 2 \sqrt{3} k + 3 - 12 = 0

{(k - \sqrt{3})}^{2} - {(2 \sqrt{3})}^{2} = 0

(k - \sqrt{3} + 2 \sqrt{3}) (k - \sqrt{3} - 2 \sqrt{3}) = 0

k + \sqrt{3} \neq 0

k = 3 \sqrt{3}

9^{{s i n}^{2} x} = 3^{\frac{3}{2}}

3^{2 {s i n}^{2} x} = 3^{\frac{3}{2}}

{s i n}^{2} x = \frac{3}{4}

s i n x = \frac{\sqrt{3}}{2} \to x = \frac{π}{3} a n d (π - \frac{π}{3}) i, e \frac{2 π}{3}

h e n c e x = \frac{π}{6}, \frac{π}{3}, \frac{2 π}{3}, \frac{5 π}{6}

s i n c e π ⩾ x ⩾ 0 s o s i n x \neq n e g e t i v e

s o s i n x \neq \frac{- \sqrt{3}}{2}

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