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

The general solution for 7sin^2(θ)-3sin(θ)=4 is θ= pi/2+2pin,θ=-0.60824…+2pin,θ=pi+0.60824…+2pin

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Popular >

7sin^2(θ)-3sin(θ)=4

Solution

7 sin^{2} (θ) - 3 sin (θ) = 4

Solution

θ = \frac{π}{2} + 2 πn, θ = - 0.60824 \dots + 2 πn, θ = π + 0.60824 \dots + 2 πn

Degrees

θ = 9 0^{\circ} + 36 0^{\circ} n, θ = - 34.84990 \dots^{\circ} + 36 0^{\circ} n, θ = 214.84990 \dots^{\circ} + 36 0^{\circ} n

Solution steps

7 sin^{2} (θ) - 3 sin (θ) = 4

Solve by substitution

7 sin^{2} (θ) - 3 sin (θ) = 4

Let:

sin (θ) = u

7 u^{2} - 3 u = 4

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

7 u^{2} - 3 u = 4

Move

4

to the left side

7 u^{2} - 3 u = 4

Subtract

4

from both sides

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

Simplify

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

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

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

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

(- 3)^{2} - 4 \cdot 7 (- 4) = 11

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

(- 3)^{2} = 3^{2}

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

Multiply the numbers:

4 \cdot 7 \cdot 4 = 112

= 3^{2} + 112

3^{2} = 9

= 9 + 112

Add the numbers:

9 + 112 = 121

= 121

Factor the number:

121 = 1 1^{2}

= 1 1^{2}

Apply radical rule:

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- ( - 3 ) \pm 11}{2 \cdot 7}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 11}{2 \cdot 7}, u_{2} = \frac{- ( - 3 ) - 11}{2 \cdot 7}

u = \frac{- ( - 3 ) + 11}{2 \cdot 7} : 1

\frac{- ( - 3 ) + 11}{2 \cdot 7}

Apply rule

- (- a) = a

= \frac{3 + 11}{2 \cdot 7}

Add the numbers:

3 + 11 = 14

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

Multiply the numbers:

2 \cdot 7 = 14

= \frac{14}{14}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- ( - 3 ) - 11}{2 \cdot 7} : - \frac{4}{7}

\frac{- ( - 3 ) - 11}{2 \cdot 7}

Apply rule

- (- a) = a

= \frac{3 - 11}{2 \cdot 7}

Subtract the numbers:

3 - 11 = - 8

= \frac{- 8}{2 \cdot 7}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{- 8}{14}

Apply the fraction rule:

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

= - \frac{8}{14}

Cancel the common factor:

2

= - \frac{4}{7}

The solutions to the quadratic equation are:

u = 1, u = - \frac{4}{7}

Substitute back

u = sin (θ)

sin (θ) = 1, sin (θ) = - \frac{4}{7}

sin (θ) = 1, sin (θ) = - \frac{4}{7}

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

sin (θ) = - \frac{4}{7} : θ = arcsin (- \frac{4}{7}) + 2 πn, θ = π + arcsin (\frac{4}{7}) + 2 πn

sin (θ) = - \frac{4}{7}

Apply trig inverse properties

sin (θ) = - \frac{4}{7}

General solutions for

sin (θ) = - \frac{4}{7}

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

θ = arcsin (- \frac{4}{7}) + 2 πn, θ = π + arcsin (\frac{4}{7}) + 2 πn

θ = arcsin (- \frac{4}{7}) + 2 πn, θ = π + arcsin (\frac{4}{7}) + 2 πn

Combine all the solutions

θ = \frac{π}{2} + 2 πn, θ = arcsin (- \frac{4}{7}) + 2 πn, θ = π + arcsin (\frac{4}{7}) + 2 πn

Show solutions in decimal form

θ = \frac{π}{2} + 2 πn, θ = - 0.60824 \dots + 2 πn, θ = π + 0.60824 \dots + 2 πn

Popular Examples

-5-4tan(θ)=-1

- 5 - 4 tan (θ) = - 1

sin(θ)=-7/10

sin (θ) = - \frac{7}{10}

10cos(2x)-3=2

10 cos (2 x) - 3 = 2

sin^2(θ)=9(cos(-θ)+1)

sin^{2} (θ) = 9 (cos (- θ) + 1)

sin(30x)=202.07

sin (30 x) = 202.07

Frequently Asked Questions (FAQ)

What is the general solution for 7sin^2(θ)-3sin(θ)=4 ?
The general solution for 7sin^2(θ)-3sin(θ)=4 is θ= pi/2+2pin,θ=-0.60824…+2pin,θ=pi+0.60824…+2pin