What is the general solution for 2cos(2θ)+5cos(θ)=4 ?

The general solution for 2cos(2θ)+5cos(θ)=4 is θ=0.72273…+2pin,θ=2pi-0.72273…+2pin

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2cos(2θ)+5cos(θ)=4

2 cos (2 θ) + 5 cos (θ) = 4

θ = 0.72273 \dots + 2 πn, θ = 2 π - 0.72273 \dots + 2 πn

Solution steps

2 cos (2 θ) + 5 cos (θ) = 4

Subtract

4

from both sides

2 cos (2 θ) + 5 cos (θ) - 4 = 0

Rewrite using trig identities

- 6 + 4 cos^{2} (θ) + 5 cos (θ) = 0

Solve by substitution

cos (θ) = \frac{3}{4}, cos (θ) = - 2

cos (θ) = \frac{3}{4} : θ = arccos (\frac{3}{4}) + 2 πn, θ = 2 π - arccos (\frac{3}{4}) + 2 πn

cos (θ) = - 2 :

No Solution

Combine all the solutions

θ = arccos (\frac{3}{4}) + 2 πn, θ = 2 π - arccos (\frac{3}{4}) + 2 πn

Show solutions in decimal form

θ = 0.72273 \dots + 2 πn, θ = 2 π - 0.72273 \dots + 2 πn

y, x = sin (\frac{1}{2} y)

sin^{2} (θ) - sin (θ) = 2

x, sin (x) cos (x) = 0

\frac{2}{3} = \frac{sin ( 13 5 ^{\circ} )}{sin ( x )}

10 = cos (x)

What is the general solution for 2cos(2θ)+5cos(θ)=4 ?
The general solution for 2cos(2θ)+5cos(θ)=4 is θ=0.72273…+2pin,θ=2pi-0.72273…+2pin