What is the general solution for -7sin^2(θ)+4sin(θ)=-3 ?

The general solution for -7sin^2(θ)+4sin(θ)=-3 is θ=-0.44291…+2pin,θ=pi+0.44291…+2pin,θ= pi/2+2pin

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Popular >

-7sin^2(θ)+4sin(θ)=-3

Solution

- 7 sin^{2} (θ) + 4 sin (θ) = - 3

Solution

θ = - 0.44291 \dots + 2 πn, θ = π + 0.44291 \dots + 2 πn, θ = \frac{π}{2} + 2 πn

Degrees

θ = - 25.37693 \dots^{\circ} + 36 0^{\circ} n, θ = 205.37693 \dots^{\circ} + 36 0^{\circ} n, θ = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

- 7 sin^{2} (θ) + 4 sin (θ) = - 3

Solve by substitution

- 7 sin^{2} (θ) + 4 sin (θ) = - 3

Let:

sin (θ) = u

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

- 7 u^{2} + 4 u = - 3 : u = - \frac{3}{7}, u = 1

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

Move

3

to the left side

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

Add

3

to both sides

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

Simplify

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 7, b = 4, c = 3

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 7 ) \cdot 3}{2 ( - 7 )}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 7 ) \cdot 3}{2 ( - 7 )}

4^{2} - 4 (- 7) \cdot 3 = 10

4^{2} - 4 (- 7) \cdot 3

Apply rule

- (- a) = a

= 4^{2} + 4 \cdot 7 \cdot 3

Multiply the numbers:

4 \cdot 7 \cdot 3 = 84

= 4^{2} + 84

4^{2} = 16

= 16 + 84

Add the numbers:

16 + 84 = 100

= 100

Factor the number:

100 = 1 0^{2}

= 1 0^{2}

Apply radical rule:

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- 4 \pm 10}{2 ( - 7 )}

Separate the solutions

u_{1} = \frac{- 4 + 10}{2 ( - 7 )}, u_{2} = \frac{- 4 - 10}{2 ( - 7 )}

u = \frac{- 4 + 10}{2 ( - 7 )} : - \frac{3}{7}

\frac{- 4 + 10}{2 ( - 7 )}

Remove parentheses:

(- a) = - a

= \frac{- 4 + 10}{- 2 \cdot 7}

Add/Subtract the numbers:

- 4 + 10 = 6

= \frac{6}{- 2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{6}{- 14}

Apply the fraction rule:

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

= - \frac{6}{14}

Cancel the common factor:

2

= - \frac{3}{7}

u = \frac{- 4 - 10}{2 ( - 7 )} : 1

\frac{- 4 - 10}{2 ( - 7 )}

Remove parentheses:

(- a) = - a

= \frac{- 4 - 10}{- 2 \cdot 7}

Subtract the numbers:

- 4 - 10 = - 14

= \frac{- 14}{- 2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{- 14}{- 14}

Apply the fraction rule:

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

= \frac{14}{14}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

u = - \frac{3}{7}, u = 1

Substitute back

u = sin (θ)

sin (θ) = - \frac{3}{7}, sin (θ) = 1

sin (θ) = - \frac{3}{7}, sin (θ) = 1

sin (θ) = - \frac{3}{7} : θ = arcsin (- \frac{3}{7}) + 2 πn, θ = π + arcsin (\frac{3}{7}) + 2 πn

sin (θ) = - \frac{3}{7}

Apply trig inverse properties

sin (θ) = - \frac{3}{7}

General solutions for

sin (θ) = - \frac{3}{7}

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

θ = arcsin (- \frac{3}{7}) + 2 πn, θ = π + arcsin (\frac{3}{7}) + 2 πn

θ = arcsin (- \frac{3}{7}) + 2 πn, θ = π + arcsin (\frac{3}{7}) + 2 πn

sin (θ) = 1 : θ = \frac{π}{2} + 2 πn

sin (θ) = 1

General solutions for

sin (θ) = 1

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 = \frac{π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn

Combine all the solutions

θ = arcsin (- \frac{3}{7}) + 2 πn, θ = π + arcsin (\frac{3}{7}) + 2 πn, θ = \frac{π}{2} + 2 πn

Show solutions in decimal form

θ = - 0.44291 \dots + 2 πn, θ = π + 0.44291 \dots + 2 πn, θ = \frac{π}{2} + 2 πn

Popular Examples

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

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

csc(θ)=-(2sqrt(3))/3

csc (θ) = - \frac{2 3}{3}

2sin(x/3)+sqrt(3)=0

2 sin (\frac{x}{3}) + 3 = 0

tan(θ)= 15/8

tan (θ) = \frac{15}{8}

sin(θ)=4

sin (θ) = 4

Frequently Asked Questions (FAQ)

What is the general solution for -7sin^2(θ)+4sin(θ)=-3 ?
The general solution for -7sin^2(θ)+4sin(θ)=-3 is θ=-0.44291…+2pin,θ=pi+0.44291…+2pin,θ= pi/2+2pin