9cos^2(θ)-10cos(θ)-2=-7cos(θ)

Solution

9 cos^{2} (θ) - 10 cos (θ) - 2 = - 7 cos (θ)

θ = 0.84106 \dots + 2 πn, θ = 2 π - 0.84106 \dots + 2 πn, θ = 1.91063 \dots + 2 πn, θ = - 1.91063 \dots + 2 πn

Solution steps

9 cos^{2} (θ) - 10 cos (θ) - 2 = - 7 cos (θ)

Solve by substitution

cos (θ) = \frac{2}{3}, cos (θ) = - \frac{1}{3}

cos (θ) = \frac{2}{3} : θ = arccos (\frac{2}{3}) + 2 πn, θ = 2 π - arccos (\frac{2}{3}) + 2 πn

cos (θ) = - \frac{1}{3} : θ = arccos (- \frac{1}{3}) + 2 πn, θ = - arccos (- \frac{1}{3}) + 2 πn

Combine all the solutions

θ = arccos (\frac{2}{3}) + 2 πn, θ = 2 π - arccos (\frac{2}{3}) + 2 πn, θ = arccos (- \frac{1}{3}) + 2 πn, θ = - arccos (- \frac{1}{3}) + 2 πn

Show solutions in decimal form

θ = 0.84106 \dots + 2 πn, θ = 2 π - 0.84106 \dots + 2 πn, θ = 1.91063 \dots + 2 πn, θ = - 1.91063 \dots + 2 πn

cos (4 x) + 1 = 0

sin (x) = - 2

4 sec (θ) - 5 = 0

3 sin (x) + 5 cos (x) = 0

sec (3 x) = - 2