2cos(θ)=3cos(3θ)

Solution

2 cos (θ) = 3 cos (3 θ)

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = 2.84874 \dots + 2 πn, θ = - 2.84874 \dots + 2 πn, θ = 0.29284 \dots + 2 πn, θ = 2 π - 0.29284 \dots + 2 πn

Solution steps

2 cos (θ) = 3 cos (3 θ)

Subtract

3 cos (3 θ)

from both sides

2 cos (θ) - 3 cos (3 θ) = 0

Rewrite using trig identities

11 cos (θ) - 12 cos^{3} (θ) = 0

Solve by substitution

cos (θ) = 0, cos (θ) = - \frac{33}{6}, cos (θ) = \frac{33}{6}

cos (θ) = 0 : θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn

cos (θ) = - \frac{33}{6} : θ = arccos (- \frac{33}{6}) + 2 πn, θ = - arccos (- \frac{33}{6}) + 2 πn

cos (θ) = \frac{33}{6} : θ = arccos (\frac{33}{6}) + 2 πn, θ = 2 π - arccos (\frac{33}{6}) + 2 πn

Combine all the solutions

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = arccos (- \frac{33}{6}) + 2 πn, θ = - arccos (- \frac{33}{6}) + 2 πn, θ = arccos (\frac{33}{6}) + 2 πn, θ = 2 π - arccos (\frac{33}{6}) + 2 πn

Show solutions in decimal form

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = 2.84874 \dots + 2 πn, θ = - 2.84874 \dots + 2 πn, θ = 0.29284 \dots + 2 πn, θ = 2 π - 0.29284 \dots + 2 πn

2 sin (\frac{x}{2}) = 2

cos (θ) = 0.7071

2 cos (x) - 5 sin (x) = 0

2 cos^{2} (x) = 3 cos (x) + 2

sin (3 0^{\circ}) = cos (2 x)