What is the general solution for -5cos(2θ)-9sin(θ)+8=-sin(θ) ?

The general solution for -5cos(2θ)-9sin(θ)+8=-sin(θ) is No Solution for θ\in\mathbb{R}

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Popular >

-5cos(2θ)-9sin(θ)+8=-sin(θ)

Solution

- 5 cos (2 θ) - 9 sin (θ) + 8 = - sin (θ)

Solution

No Solution for θ \in R

Solution steps

- 5 cos (2 θ) - 9 sin (θ) + 8 = - sin (θ)

Subtract

- sin (θ)

from both sides

- 5 cos (2 θ) - 8 sin (θ) + 8 = 0

Rewrite using trig identities

8 - 5 cos (2 θ) - 8 sin (θ)

Use the Double Angle identity:

cos (2 x) = 1 - 2 sin^{2} (x)

= 8 - 5 (1 - 2 sin^{2} (θ)) - 8 sin (θ)

Simplify

8 - 5 (1 - 2 sin^{2} (θ)) - 8 sin (θ) : 10 sin^{2} (θ) - 8 sin (θ) + 3

8 - 5 (1 - 2 sin^{2} (θ)) - 8 sin (θ)

Expand

- 5 (1 - 2 sin^{2} (θ)) : - 5 + 10 sin^{2} (θ)

- 5 (1 - 2 sin^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = - 5, b = 1, c = 2 sin^{2} (θ)

= - 5 \cdot 1 - (- 5) \cdot 2 sin^{2} (θ)

Apply minus-plus rules

- (- a) = a

= - 5 \cdot 1 + 5 \cdot 2 sin^{2} (θ)

Simplify

- 5 \cdot 1 + 5 \cdot 2 sin^{2} (θ) : - 5 + 10 sin^{2} (θ)

- 5 \cdot 1 + 5 \cdot 2 sin^{2} (θ)

Multiply the numbers:

5 \cdot 1 = 5

= - 5 + 5 \cdot 2 sin^{2} (θ)

Multiply the numbers:

5 \cdot 2 = 10

= - 5 + 10 sin^{2} (θ)

= - 5 + 10 sin^{2} (θ)

= 8 - 5 + 10 sin^{2} (θ) - 8 sin (θ)

Subtract the numbers:

8 - 5 = 3

= 10 sin^{2} (θ) - 8 sin (θ) + 3

= 10 sin^{2} (θ) - 8 sin (θ) + 3

3 + 10 sin^{2} (θ) - 8 sin (θ) = 0

Solve by substitution

3 + 10 sin^{2} (θ) - 8 sin (θ) = 0

Let:

sin (θ) = u

3 + 10 u^{2} - 8 u = 0

3 + 10 u^{2} - 8 u = 0 : u = \frac{2}{5} + i \frac{14}{10}, u = \frac{2}{5} - i \frac{14}{10}

3 + 10 u^{2} - 8 u = 0

Write in the standard form

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

10 u^{2} - 8 u + 3 = 0

Solve with the quadratic formula

10 u^{2} - 8 u + 3 = 0

Quadratic Equation Formula:

For

a = 10, b = - 8, c = 3

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 10 \cdot 3}{2 \cdot 10}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 10 \cdot 3}{2 \cdot 10}

Simplify

(- 8)^{2} - 4 \cdot 10 \cdot 3 : 214 i

(- 8)^{2} - 4 \cdot 10 \cdot 3

Apply exponent rule:

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

n

is even

(- 8)^{2} = 8^{2}

= 8^{2} - 4 \cdot 10 \cdot 3

Multiply the numbers:

4 \cdot 10 \cdot 3 = 120

= 8^{2} - 120

Apply imaginary number rule:

- a = i a

= i 120 - 8^{2}

- 8^{2} + 120 = 214

- 8^{2} + 120

8^{2} = 64

= - 64 + 120

Add/Subtract the numbers:

- 64 + 120 = 56

= 56

Prime factorization of

56 : 2^{3} \cdot 7

56

56

divides by

256 = 28 \cdot 2

= 2 \cdot 28

28

divides by

228 = 14 \cdot 2

= 2 \cdot 2 \cdot 14

14

divides by

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 7

2, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 7

= 2^{3} \cdot 7

= 2^{3} \cdot 7

Apply exponent rule:

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

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

Apply radical rule:

= 2^{2} 2 \cdot 7

Apply radical rule:

2^{2} = 2

= 2 2 \cdot 7

Refine

= 214

= 214 i

u_{1, 2} = \frac{- ( - 8 ) \pm 2 14 i}{2 \cdot 10}

Separate the solutions

u_{1} = \frac{- ( - 8 ) + 2 14 i}{2 \cdot 10}, u_{2} = \frac{- ( - 8 ) - 2 14 i}{2 \cdot 10}

u = \frac{- ( - 8 ) + 2 14 i}{2 \cdot 10} : \frac{2}{5} + i \frac{14}{10}

\frac{- ( - 8 ) + 2 14 i}{2 \cdot 10}

Apply rule

- (- a) = a

= \frac{8 + 2 14 i}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{8 + 2 14 i}{20}

Factor

8 + 214 i : 2 (4 + 14 i)

8 + 214 i

Rewrite as

= 2 \cdot 4 + 214 i

Factor out common term

2

= 2 (4 + 14 i)

= \frac{2 ( 4 + 14 i )}{20}

Cancel the common factor:

2

= \frac{4 + 14 i}{10}

Rewrite

\frac{4 + 14 i}{10}

in standard complex form:

\frac{2}{5} + \frac{14}{10} i

\frac{4 + 14 i}{10}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{4 + 14 i}{10} = \frac{4}{10} + \frac{14 i}{10}

= \frac{4}{10} + \frac{14 i}{10}

Cancel

\frac{4}{10} : \frac{2}{5}

\frac{4}{10}

Cancel the common factor:

2

= \frac{2}{5}

= \frac{2}{5} + \frac{14 i}{10}

Cancel

\frac{14 i}{10} : \frac{7 i}{5 2}

\frac{14 i}{10}

Factor

14 : 27

Factor

14 = 2 \cdot 7

= 2 \cdot 7

Apply radical rule:

= 27

Factor

10 : 2 \cdot 5

Factor

10 = 2 \cdot 5

= \frac{2 7 i}{2 \cdot 5}

Cancel

\frac{2 7 i}{2 \cdot 5} : \frac{7 i}{2 \cdot 5}

\frac{2 7 i}{2 \cdot 5}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}} 7 i}{2 \cdot 5}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{7 i}{5 \cdot 2 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

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

= \frac{7 i}{5 \cdot 2 ^{\frac{1}{2}}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{7 i}{5 2}

= \frac{7 i}{2 \cdot 5}

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

\frac{7}{5 2} = \frac{14}{10}

\frac{7}{5 2}

Multiply by the conjugate

\frac{2}{2}

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

72 = 14

72

Apply radical rule:

a b = a \cdot b

72 = 7 \cdot 2

= 7 \cdot 2

Multiply the numbers:

7 \cdot 2 = 14

= 14

522 = 10

522

Apply radical rule:

a a = a

22 = 2

= 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 10

= \frac{14}{10}

= \frac{2}{5} + \frac{14}{10} i

= \frac{2}{5} + \frac{14}{10} i

u = \frac{- ( - 8 ) - 2 14 i}{2 \cdot 10} : \frac{2}{5} - i \frac{14}{10}

\frac{- ( - 8 ) - 2 14 i}{2 \cdot 10}

Apply rule

- (- a) = a

= \frac{8 - 2 14 i}{2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{8 - 2 14 i}{20}

Factor

8 - 214 i : 2 (4 - 14 i)

8 - 214 i

Rewrite as

= 2 \cdot 4 - 214 i

Factor out common term

2

= 2 (4 - 14 i)

= \frac{2 ( 4 - 14 i )}{20}

Cancel the common factor:

2

= \frac{4 - 14 i}{10}

Rewrite

\frac{4 - 14 i}{10}

in standard complex form:

\frac{2}{5} - \frac{14}{10} i

\frac{4 - 14 i}{10}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{4 - 14 i}{10} = \frac{4}{10} - \frac{14 i}{10}

= \frac{4}{10} - \frac{14 i}{10}

Cancel

\frac{4}{10} : \frac{2}{5}

\frac{4}{10}

Cancel the common factor:

2

= \frac{2}{5}

= \frac{2}{5} - \frac{14 i}{10}

Cancel

\frac{14 i}{10} : \frac{7 i}{5 2}

\frac{14 i}{10}

Factor

14 : 27

Factor

14 = 2 \cdot 7

= 2 \cdot 7

Apply radical rule:

= 27

Factor

10 : 2 \cdot 5

Factor

10 = 2 \cdot 5

= \frac{2 7 i}{2 \cdot 5}

Cancel

\frac{2 7 i}{2 \cdot 5} : \frac{7 i}{2 \cdot 5}

\frac{2 7 i}{2 \cdot 5}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}} 7 i}{2 \cdot 5}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{7 i}{5 \cdot 2 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

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

= \frac{7 i}{5 \cdot 2 ^{\frac{1}{2}}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{7 i}{5 2}

= \frac{7 i}{2 \cdot 5}

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

- \frac{7}{5 2} = - \frac{14}{10}

- \frac{7}{5 2}

Multiply by the conjugate

\frac{2}{2}

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

72 = 14

72

Apply radical rule:

a b = a \cdot b

72 = 7 \cdot 2

= 7 \cdot 2

Multiply the numbers:

7 \cdot 2 = 14

= 14

522 = 10

522

Apply radical rule:

a a = a

22 = 2

= 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 10

= - \frac{14}{10}

= \frac{2}{5} - \frac{14}{10} i

= \frac{2}{5} - \frac{14}{10} i

The solutions to the quadratic equation are:

u = \frac{2}{5} + i \frac{14}{10}, u = \frac{2}{5} - i \frac{14}{10}

Substitute back

u = sin (θ)

sin (θ) = \frac{2}{5} + i \frac{14}{10}, sin (θ) = \frac{2}{5} - i \frac{14}{10}

sin (θ) = \frac{2}{5} + i \frac{14}{10}, sin (θ) = \frac{2}{5} - i \frac{14}{10}

sin (θ) = \frac{2}{5} + i \frac{14}{10} :

No Solution

sin (θ) = \frac{2}{5} + i \frac{14}{10}

No Solution

sin (θ) = \frac{2}{5} - i \frac{14}{10} :

No Solution

sin (θ) = \frac{2}{5} - i \frac{14}{10}

No Solution

Combine all the solutions

No Solution for θ \in R

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Frequently Asked Questions (FAQ)

What is the general solution for -5cos(2θ)-9sin(θ)+8=-sin(θ) ?
The general solution for -5cos(2θ)-9sin(θ)+8=-sin(θ) is No Solution for θ\in\mathbb{R}