What is the general solution for 2sin(2θ)-3sin(θ)=0 ?

The general solution for 2sin(2θ)-3sin(θ)=0 is θ=2pin,θ=pi+2pin,θ=0.72273…+2pin,θ=2pi-0.72273…+2pin

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Popular >

2sin(2θ)-3sin(θ)=0

Solution

2 sin (2 θ) - 3 sin (θ) = 0

Solution

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

Degrees

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n, θ = 41.40962 \dots^{\circ} + 36 0^{\circ} n, θ = 318.59037 \dots^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (2 θ) - 3 sin (θ) = 0

Rewrite using trig identities

2 sin (2 θ) - 3 sin (θ)

Use the Double Angle identity:

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

= 2 \cdot 2 sin (θ) cos (θ) - 3 sin (θ)

Simplify

= 4 sin (θ) cos (θ) - 3 sin (θ)

- 3 sin (θ) + 4 cos (θ) sin (θ) = 0

Factor

- 3 sin (θ) + 4 cos (θ) sin (θ) : sin (θ) (4 cos (θ) - 3)

- 3 sin (θ) + 4 cos (θ) sin (θ)

Factor out common term

sin (θ)

= sin (θ) (- 3 + 4 cos (θ))

sin (θ) (4 cos (θ) - 3) = 0

Solving each part separately

sin (θ) = 0 or 4 cos (θ) - 3 = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

General solutions for

sin (θ) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

4 cos (θ) - 3 = 0 : θ = arccos (\frac{3}{4}) + 2 πn, θ = 2 π - arccos (\frac{3}{4}) + 2 πn

4 cos (θ) - 3 = 0

Move

3

to the right side

4 cos (θ) - 3 = 0

Add

3

to both sides

4 cos (θ) - 3 + 3 = 0 + 3

Simplify

4 cos (θ) = 3

4 cos (θ) = 3

Divide both sides by

4

4 cos (θ) = 3

Divide both sides by

4

\frac{4 cos ( θ )}{4} = \frac{3}{4}

Simplify

cos (θ) = \frac{3}{4}

cos (θ) = \frac{3}{4}

Apply trig inverse properties

cos (θ) = \frac{3}{4}

General solutions for

cos (θ) = \frac{3}{4}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

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

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

Combine all the solutions

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

Show solutions in decimal form

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

Popular Examples

cos^2(x)+cos^2(x/2)-1/2 =0

cos^{2} (x) + cos^{2} (\frac{x}{2}) - \frac{1}{2} = 0

4cos^2(x)-1=0,0<= x<= 2pi

4 cos^{2} (x) - 1 = 0, 0 \leq x \leq 2 π

cos(2x)+sqrt(3)sin(x)=1

cos (2 x) + 3 sin (x) = 1

tan(2θ)-2cos(θ)=0

tan (2 θ) - 2 cos (θ) = 0

tan(θ)= 12/5 ,pi<θ<(3pi)/2

tan (θ) = \frac{12}{5}, π < θ < \frac{3 π}{2}

Frequently Asked Questions (FAQ)

What is the general solution for 2sin(2θ)-3sin(θ)=0 ?
The general solution for 2sin(2θ)-3sin(θ)=0 is θ=2pin,θ=pi+2pin,θ=0.72273…+2pin,θ=2pi-0.72273…+2pin