What is the general solution for 3sin^2(θ)+3sin(θ)+3=7sin(θ)+7 ?

The general solution for 3sin^2(θ)+3sin(θ)+3=7sin(θ)+7 is θ=-0.72972…+2pin,θ=pi+0.72972…+2pin

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Popular >

3sin^2(θ)+3sin(θ)+3=7sin(θ)+7

Solution

3 sin^{2} (θ) + 3 sin (θ) + 3 = 7 sin (θ) + 7

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 = 221.81031 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 sin^{2} (θ) + 3 sin (θ) + 3 = 7 sin (θ) + 7

Solve by substitution

3 sin^{2} (θ) + 3 sin (θ) + 3 = 7 sin (θ) + 7

Let:

sin (θ) = u

3 u^{2} + 3 u + 3 = 7 u + 7

3 u^{2} + 3 u + 3 = 7 u + 7 : u = 2, u = - \frac{2}{3}

3 u^{2} + 3 u + 3 = 7 u + 7

Move

7

to the left side

3 u^{2} + 3 u + 3 = 7 u + 7

Subtract

7

from both sides

3 u^{2} + 3 u + 3 - 7 = 7 u + 7 - 7

Simplify

3 u^{2} + 3 u - 4 = 7 u

3 u^{2} + 3 u - 4 = 7 u

Move

7 u

to the left side

3 u^{2} + 3 u - 4 = 7 u

Subtract

7 u

from both sides

3 u^{2} + 3 u - 4 - 7 u = 7 u - 7 u

Simplify

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 4)^{2} = 4^{2}

= 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} : 2

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

Apply rule

- (- a) = a

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

Add the numbers:

4 + 8 = 12

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= 2

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

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

Apply rule

- (- a) = a

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

Subtract the numbers:

4 - 8 = - 4

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 4}{6}

Apply the fraction rule:

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

= - \frac{4}{6}

Cancel the common factor:

2

= - \frac{2}{3}

The solutions to the quadratic equation are:

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

Substitute back

u = sin (θ)

sin (θ) = 2, sin (θ) = - \frac{2}{3}

sin (θ) = 2, sin (θ) = - \frac{2}{3}

sin (θ) = 2 :

No Solution

sin (θ) = 2

- 1 \leq sin (x) \leq 1

No Solution

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

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

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Frequently Asked Questions (FAQ)

What is the general solution for 3sin^2(θ)+3sin(θ)+3=7sin(θ)+7 ?
The general solution for 3sin^2(θ)+3sin(θ)+3=7sin(θ)+7 is θ=-0.72972…+2pin,θ=pi+0.72972…+2pin