cos^2(x)=8sin^2(x)-6sin(x)

Solution

cos^{2} (x) = 8 sin^{2} (x) - 6 sin (x)

x = - 0.13851 \dots + 2 πn, x = π + 0.13851 \dots + 2 πn, x = 0.93523 \dots + 2 πn, x = π - 0.93523 \dots + 2 πn

Solution steps

cos^{2} (x) = 8 sin^{2} (x) - 6 sin (x)

Subtract

8 sin^{2} (x) - 6 sin (x)

from both sides

cos^{2} (x) - 8 sin^{2} (x) + 6 sin (x) = 0

Rewrite using trig identities

1 + 6 sin (x) - 9 sin^{2} (x) = 0

Solve by substitution

sin (x) = - \frac{- 1 + 2}{3}, sin (x) = \frac{1 + 2}{3}

sin (x) = - \frac{- 1 + 2}{3} : x = arcsin (- \frac{- 1 + 2}{3}) + 2 πn, x = π + arcsin (\frac{- 1 + 2}{3}) + 2 πn

sin (x) = \frac{1 + 2}{3} : x = arcsin (\frac{1 + 2}{3}) + 2 πn, x = π - arcsin (\frac{1 + 2}{3}) + 2 πn

Combine all the solutions

x = arcsin (- \frac{- 1 + 2}{3}) + 2 πn, x = π + arcsin (\frac{- 1 + 2}{3}) + 2 πn, x = arcsin (\frac{1 + 2}{3}) + 2 πn, x = π - arcsin (\frac{1 + 2}{3}) + 2 πn

Show solutions in decimal form

x = - 0.13851 \dots + 2 πn, x = π + 0.13851 \dots + 2 πn, x = 0.93523 \dots + 2 πn, x = π - 0.93523 \dots + 2 πn

3 sin (x) - cos (x) = 1

cos (x) sin (x) = - 2 sin (x)

(csc (θ) - 2) (cot (θ) + 1) = 0

sin (x) = csc (x)

9 sin (x) tan (x) = 10 tan (x)