cos(2x)-2sin^2(x)-sin(x)=1

Solution

cos (2 x) - 2 sin^{2} (x) - sin (x) = 1

x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn, x = 2 πn, x = π + 2 πn

Solution steps

cos (2 x) - 2 sin^{2} (x) - sin (x) = 1

Subtract

1

from both sides

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

Rewrite using trig identities

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

Solve by substitution

sin (x) = - \frac{1}{4}, sin (x) = 0

sin (x) = - \frac{1}{4} : x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

sin (x) = 0 : x = 2 πn, x = π + 2 πn

Combine all the solutions

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn, x = 2 πn, x = π + 2 πn

Show solutions in decimal form

x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn, x = 2 πn, x = π + 2 πn

- 8 sin (θ) = - 3

sin (x) + 1 = 1

- 12 cot (x) - 43 = 0

3 sin^{2} (t) = 2 cos (t) sin (t)

sin (x + \frac{π}{4}) + sin (x - \frac{π}{4}) = 0