9cos(2x)=9cos^2(x)-4

Solution

9 cos (2 x) = 9 cos^{2} (x) - 4

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn, x = - 0.72972 \dots + 2 πn, x = π + 0.72972 \dots + 2 πn

Solution steps

9 cos (2 x) = 9 cos^{2} (x) - 4

Subtract

9 cos^{2} (x) - 4

from both sides

9 cos (2 x) - 9 cos^{2} (x) + 4 = 0

Rewrite using trig identities

4 - 9 sin^{2} (x) = 0

Solve by substitution

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

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

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

Combine all the solutions

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

Show solutions in decimal form

x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn, x = - 0.72972 \dots + 2 πn, x = π + 0.72972 \dots + 2 πn

2 - cos (x) = 0

sin (\frac{1}{x}) = 1

2 sin (θ) - cos (θ) = 0

cot (\frac{x}{7}) = 12

sin (θ) = \frac{4}{9}