log_{8}(4)=((sin^2(x)))/(sin(x)-1/8)

Solution

lo g_{8} (4) = \frac{( sin ^{2} ( x ) )}{sin ( x ) - \frac{1}{8}}

x = 0.16744 \dots + 2 πn, x = π - 0.16744 \dots + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

lo g_{8} (4) = \frac{( sin ^{2} ( x ) )}{sin ( x ) - \frac{1}{8}}

Subtract

\frac{sin ^{2} ( x )}{sin ( x ) - \frac{1}{8}}

from both sides

\frac{2}{3} - \frac{8 sin ^{2} ( x )}{8 sin ( x ) - 1} = 0

Simplify

\frac{2}{3} - \frac{8 sin ^{2} ( x )}{8 sin ( x ) - 1} : \frac{2 ( 8 sin ( x ) - 1 ) - 24 sin ^{2} ( x )}{3 ( 8 sin ( x ) - 1 )}

\frac{2 ( 8 sin ( x ) - 1 ) - 24 sin ^{2} ( x )}{3 ( 8 sin ( x ) - 1 )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 (8 sin (x) - 1) - 24 sin^{2} (x) = 0

Solve by substitution

sin (x) = \frac{1}{6}, sin (x) = \frac{1}{2}

sin (x) = \frac{1}{6} : x = arcsin (\frac{1}{6}) + 2 πn, x = π - arcsin (\frac{1}{6}) + 2 πn

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = arcsin (\frac{1}{6}) + 2 πn, x = π - arcsin (\frac{1}{6}) + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Show solutions in decimal form

x = 0.16744 \dots + 2 πn, x = π - 0.16744 \dots + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

4 \cdot cos^{2} (x) - 1 = 0

tan (2 x - 5) = sin^{2} (3 0^{\circ}) + sin^{2} (6 0^{\circ})

5 sin (x) (sin (x) - 1) = 0

5 tan^{2} (x) - 9 sec (x) + 2 = 0

- sin (A) + 2 cos (A) sin (A) = 0