What is the general solution for 3sin^2(θ)-4=-4sin(θ) ?

The general solution for 3sin^2(θ)-4=-4sin(θ) is θ=0.72972…+2pin,θ=pi-0.72972…+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

3sin^2(θ)-4=-4sin(θ)

Solution

3 sin^{2} (θ) - 4 = - 4 sin (θ)

Solution

θ = 0.72972 \dots + 2 πn, θ = π - 0.72972 \dots + 2 πn

Degrees

θ = 41.81031 \dots^{\circ} + 36 0^{\circ} n, θ = 138.18968 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 sin^{2} (θ) - 4 = - 4 sin (θ)

Solve by substitution

3 sin^{2} (θ) - 4 = - 4 sin (θ)

Let:

sin (θ) = u

3 u^{2} - 4 = - 4 u

3 u^{2} - 4 = - 4 u : u = \frac{2}{3}, u = - 2

3 u^{2} - 4 = - 4 u

Move

4 u

to the left side

3 u^{2} - 4 = - 4 u

Add

4 u

to both sides

3 u^{2} - 4 + 4 u = - 4 u + 4 u

Simplify

3 u^{2} - 4 + 4 u = 0

3 u^{2} - 4 + 4 u = 0

Write in the standard form

a x^{2} + b x + c = 0

3 u^{2} + 4 u - 4 = 0

Solve with the quadratic formula

3 u^{2} + 4 u - 4 = 0

Quadratic Equation Formula:

For

a = 3, b = 4, c = - 4

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 3 ( - 4 )}{2 \cdot 3}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 3 ( - 4 )}{2 \cdot 3}

4^{2} - 4 \cdot 3 (- 4) = 8

4^{2} - 4 \cdot 3 (- 4)

Apply rule

- (- a) = a

= 4^{2} + 4 \cdot 3 \cdot 4

Multiply the numbers:

4 \cdot 3 \cdot 4 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

Add the numbers:

16 + 48 = 64

= 64

Factor the number:

64 = 8^{2}

= 8^{2}

Apply radical rule:

8^{2} = 8

= 8

u_{1, 2} = \frac{- 4 \pm 8}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- 4 + 8}{2 \cdot 3}, u_{2} = \frac{- 4 - 8}{2 \cdot 3}

u = \frac{- 4 + 8}{2 \cdot 3} : \frac{2}{3}

\frac{- 4 + 8}{2 \cdot 3}

Add/Subtract the numbers:

- 4 + 8 = 4

= \frac{4}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{4}{6}

Cancel the common factor:

2

= \frac{2}{3}

u = \frac{- 4 - 8}{2 \cdot 3} : - 2

\frac{- 4 - 8}{2 \cdot 3}

Subtract the numbers:

- 4 - 8 = - 12

= \frac{- 12}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 12}{6}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= - 2

The solutions to the quadratic equation are:

u = \frac{2}{3}, u = - 2

Substitute back

u = sin (θ)

sin (θ) = \frac{2}{3}, sin (θ) = - 2

sin (θ) = \frac{2}{3}, sin (θ) = - 2

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

sin (θ) = \frac{2}{3}

Apply trig inverse properties

sin (θ) = \frac{2}{3}

General solutions for

sin (θ) = \frac{2}{3}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

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

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

sin (θ) = - 2 :

No Solution

sin (θ) = - 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

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

Show solutions in decimal form

θ = 0.72972 \dots + 2 πn, θ = π - 0.72972 \dots + 2 πn

Popular Examples

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

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

4cot(θ)sin(θ)=2

4 cot (θ) sin (θ) = 2

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

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

75= 120/4*cos((2*pi*x)/(360))+120/2

75 = \frac{120}{4} \cdot cos (\frac{2 \cdot π \cdot x}{360}) + \frac{120}{2}

cot^2(x)+2=2csc(x)

cot^{2} (x) + 2 = 2 csc (x)

Frequently Asked Questions (FAQ)

What is the general solution for 3sin^2(θ)-4=-4sin(θ) ?
The general solution for 3sin^2(θ)-4=-4sin(θ) is θ=0.72972…+2pin,θ=pi-0.72972…+2pin