What is the general solution for -7sin^2(θ)+2sin(θ)+7=0 ?

The general solution for -7sin^2(θ)+2sin(θ)+7=0 is θ=-1.04974…+2pin,θ=pi+1.04974…+2pin

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Popular >

-7sin^2(θ)+2sin(θ)+7=0

Solution

- 7 sin^{2} (θ) + 2 sin (θ) + 7 = 0

Solution

θ = - 1.04974 \dots + 2 πn, θ = π + 1.04974 \dots + 2 πn

Degrees

θ = - 60.14585 \dots^{\circ} + 36 0^{\circ} n, θ = 240.14585 \dots^{\circ} + 36 0^{\circ} n

Solution steps

- 7 sin^{2} (θ) + 2 sin (θ) + 7 = 0

Solve by substitution

- 7 sin^{2} (θ) + 2 sin (θ) + 7 = 0

Let:

sin (θ) = u

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

- 7 u^{2} + 2 u + 7 = 0 : u = - \frac{- 1 + 5 2}{7}, u = \frac{1 + 5 2}{7}

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 7, b = 2, c = 7

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

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

2^{2} - 4 (- 7) \cdot 7 = 102

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

Apply rule

- (- a) = a

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

Multiply the numbers:

4 \cdot 7 \cdot 7 = 196

= 2^{2} + 196

2^{2} = 4

= 4 + 196

Add the numbers:

4 + 196 = 200

= 200

Prime factorization of

200 : 2^{3} \cdot 5^{2}

200

200

divides by

2200 = 100 \cdot 2

= 2 \cdot 100

100

divides by

2100 = 50 \cdot 2

= 2 \cdot 2 \cdot 50

50

divides by

250 = 25 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 25

25

divides by

525 = 5 \cdot 5

= 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5

= 2^{3} \cdot 5^{2}

= 2^{3} \cdot 5^{2}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 5^{2} \cdot 2

Apply radical rule:

= 2 2^{2} 5^{2}

Apply radical rule:

2^{2} = 2

= 22 5^{2}

Apply radical rule:

5^{2} = 5

= 2 \cdot 52

Refine

= 102

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

Separate the solutions

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

u = \frac{- 2 + 10 2}{2 ( - 7 )} : - \frac{- 1 + 5 2}{7}

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

Remove parentheses:

(- a) = - a

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

Multiply the numbers:

2 \cdot 7 = 14

= \frac{- 2 + 10 2}{- 14}

Apply the fraction rule:

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

= - \frac{- 2 + 10 2}{14}

Cancel

\frac{- 2 + 10 2}{14} : \frac{5 2 - 1}{7}

\frac{- 2 + 10 2}{14}

Factor

- 2 + 102 : 2 (- 1 + 52)

- 2 + 102

Rewrite as

= - 2 \cdot 1 + 2 \cdot 52

Factor out common term

2

= 2 (- 1 + 52)

= \frac{2 ( - 1 + 5 2 )}{14}

Cancel the common factor:

2

= \frac{- 1 + 5 2}{7}

= - \frac{5 2 - 1}{7}

= - \frac{- 1 + 5 2}{7}

u = \frac{- 2 - 10 2}{2 ( - 7 )} : \frac{1 + 5 2}{7}

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

Remove parentheses:

(- a) = - a

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

Multiply the numbers:

2 \cdot 7 = 14

= \frac{- 2 - 10 2}{- 14}

Apply the fraction rule:

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

- 2 - 102 = - (2 + 102)

= \frac{2 + 10 2}{14}

Factor

2 + 102 : 2 (1 + 52)

2 + 102

Rewrite as

= 2 \cdot 1 + 2 \cdot 52

Factor out common term

2

= 2 (1 + 52)

= \frac{2 ( 1 + 5 2 )}{14}

Cancel the common factor:

2

= \frac{1 + 5 2}{7}

The solutions to the quadratic equation are:

u = - \frac{- 1 + 5 2}{7}, u = \frac{1 + 5 2}{7}

Substitute back

u = sin (θ)

sin (θ) = - \frac{- 1 + 5 2}{7}, sin (θ) = \frac{1 + 5 2}{7}

sin (θ) = - \frac{- 1 + 5 2}{7}, sin (θ) = \frac{1 + 5 2}{7}

sin (θ) = - \frac{- 1 + 5 2}{7} : θ = arcsin (- \frac{- 1 + 5 2}{7}) + 2 πn, θ = π + arcsin (\frac{- 1 + 5 2}{7}) + 2 πn

sin (θ) = - \frac{- 1 + 5 2}{7}

Apply trig inverse properties

sin (θ) = - \frac{- 1 + 5 2}{7}

General solutions for

sin (θ) = - \frac{- 1 + 5 2}{7}

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

θ = arcsin (- \frac{- 1 + 5 2}{7}) + 2 πn, θ = π + arcsin (\frac{- 1 + 5 2}{7}) + 2 πn

θ = arcsin (- \frac{- 1 + 5 2}{7}) + 2 πn, θ = π + arcsin (\frac{- 1 + 5 2}{7}) + 2 πn

sin (θ) = \frac{1 + 5 2}{7} :

No Solution

sin (θ) = \frac{1 + 5 2}{7}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

θ = arcsin (- \frac{- 1 + 5 2}{7}) + 2 πn, θ = π + arcsin (\frac{- 1 + 5 2}{7}) + 2 πn

Show solutions in decimal form

θ = - 1.04974 \dots + 2 πn, θ = π + 1.04974 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for -7sin^2(θ)+2sin(θ)+7=0 ?
The general solution for -7sin^2(θ)+2sin(θ)+7=0 is θ=-1.04974…+2pin,θ=pi+1.04974…+2pin