2cos(2x)-sin(x)-1=0

Solution

2 cos (2 x) - sin (x) - 1 = 0

x = - 0.69500 \dots + 2 πn, x = π + 0.69500 \dots + 2 πn, x = 0.40105 \dots + 2 πn, x = π - 0.40105 \dots + 2 πn

Solution steps

2 cos (2 x) - sin (x) - 1 = 0

Rewrite using trig identities

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

Solve by substitution

sin (x) = - \frac{1 + 17}{8}, sin (x) = \frac{17 - 1}{8}

sin (x) = - \frac{1 + 17}{8} : x = arcsin (- \frac{1 + 17}{8}) + 2 πn, x = π + arcsin (\frac{1 + 17}{8}) + 2 πn

sin (x) = \frac{17 - 1}{8} : x = arcsin (\frac{17 - 1}{8}) + 2 πn, x = π - arcsin (\frac{17 - 1}{8}) + 2 πn

Combine all the solutions

x = arcsin (- \frac{1 + 17}{8}) + 2 πn, x = π + arcsin (\frac{1 + 17}{8}) + 2 πn, x = arcsin (\frac{17 - 1}{8}) + 2 πn, x = π - arcsin (\frac{17 - 1}{8}) + 2 πn

Show solutions in decimal form

x = - 0.69500 \dots + 2 πn, x = π + 0.69500 \dots + 2 πn, x = 0.40105 \dots + 2 πn, x = π - 0.40105 \dots + 2 πn

2 cos^{2} (θ) - 2 sin^{2} (θ) = 2

cot (θ) + 3 csc (θ) = 5

2 cos^{2} (θ) - 2 sin^{2} (θ) = 3

1 - ∣ cos (x) ∣ = 0

sin (θ) = - 0.789