What is the general solution for 2sin(θ)=3(1-cos(θ)) ?

The general solution for 2sin(θ)=3(1-cos(θ)) is θ=1.17600…+2pin,θ=2pin

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2sin(θ)=3(1-cos(θ))

2 sin (θ) = 3 (1 - cos (θ))

θ = 1.17600 \dots + 2 πn, θ = 2 πn

Solution steps

2 sin (θ) = 3 (1 - cos (θ))

Square both sides

(2 sin (θ))^{2} = (3 (1 - cos (θ)))^{2}

Subtract

(3 (1 - cos (θ)))^{2}

from both sides

4 sin^{2} (θ) - 9 (1 - cos (θ))^{2} = 0

Rewrite using trig identities

- 5 - 13 cos^{2} (θ) + 18 cos (θ) = 0

Solve by substitution

cos (θ) = \frac{5}{13}, cos (θ) = 1

cos (θ) = \frac{5}{13} : θ = arccos (\frac{5}{13}) + 2 πn, θ = 2 π - arccos (\frac{5}{13}) + 2 πn

cos (θ) = 1 : θ = 2 πn

Combine all the solutions

θ = arccos (\frac{5}{13}) + 2 πn, θ = 2 π - arccos (\frac{5}{13}) + 2 πn, θ = 2 πn

Verify solutions by plugging them into the original equation

θ = arccos (\frac{5}{13}) + 2 πn, θ = 2 πn

Show solutions in decimal form

θ = 1.17600 \dots + 2 πn, θ = 2 πn

4 sin^{2} (x) tan (x) - 4 sin^{2} (x) - 3 tan (x) + 3 = 0

cos (t) = - 0, 5

sinh (x) = sinh (- x)

\frac{π}{4} = arctan (t)

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

What is the general solution for 2sin(θ)=3(1-cos(θ)) ?
The general solution for 2sin(θ)=3(1-cos(θ)) is θ=1.17600…+2pin,θ=2pin