What is the general solution for [sin(x)+1/2 ]^2= 1/2 sin(x)+13/16 ?

The general solution for [sin(x)+1/2 ]^2= 1/2 sin(x)+13/16 is x=0.57111…+2pin,x=pi-0.57111…+2pin

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[sin(x)+1/2 ]^2= 1/2 sin(x)+13/16

Solution

[sin (x) + \frac{1}{2}]^{2} = \frac{1}{2} sin (x) + \frac{13}{16}

Solution

x = 0.57111 \dots + 2 πn, x = π - 0.57111 \dots + 2 πn

Degrees

x = 32.72240 \dots^{\circ} + 36 0^{\circ} n, x = 147.27759 \dots^{\circ} + 36 0^{\circ} n

Solution steps

[sin (x) + \frac{1}{2}]^{2} = \frac{1}{2} sin (x) + \frac{13}{16}

Solve by substitution

[sin (x) + \frac{1}{2}]^{2} = \frac{1}{2} sin (x) + \frac{13}{16}

Let:

sin (x) = u

[u + \frac{1}{2}]^{2} = \frac{1}{2} u + \frac{13}{16}

[u + \frac{1}{2}]^{2} = \frac{1}{2} u + \frac{13}{16} : u = \frac{- 1 + 10}{4}, u = - \frac{1 + 10}{4}

[u + \frac{1}{2}]^{2} = \frac{1}{2} u + \frac{13}{16}

Find Least Common Multiplier of

2, 16 : 16

2, 16

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

16 : 2 \cdot 2 \cdot 2 \cdot 2

16

16

divides by

216 = 8 \cdot 2

= 2 \cdot 8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

16

= 2 \cdot 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 \cdot 2 = 16

= 16

Multiply by LCM=

16

[u + \frac{1}{2}]^{2} \cdot 16 = \frac{1}{2} u \cdot 16 + \frac{13}{16} \cdot 16

Simplify

16 (u + \frac{1}{2})^{2} = 8 u + 13

Expand

16 (u + \frac{1}{2})^{2} : 16 u^{2} + 16 u + 4

16 (u + \frac{1}{2})^{2}

(u + \frac{1}{2})^{2} = u^{2} + u + \frac{1}{4}

(u + \frac{1}{2})^{2}

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = u, b = \frac{1}{2}

= u^{2} + 2 u \frac{1}{2} + (\frac{1}{2})^{2}

Simplify

u^{2} + 2 u \frac{1}{2} + (\frac{1}{2})^{2} : u^{2} + u + \frac{1}{4}

u^{2} + 2 u \frac{1}{2} + (\frac{1}{2})^{2}

2 u \frac{1}{2} = u

2 u \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 2}{2} u

Cancel the common factor:

2

= u \cdot 1

Multiply:

u \cdot 1 = u

= u

(\frac{1}{2})^{2} = \frac{1}{4}

(\frac{1}{2})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Apply rule

1^{a} = 1

1^{2} = 1

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

2^{2} = 4

= \frac{1}{4}

= u^{2} + u + \frac{1}{4}

= u^{2} + u + \frac{1}{4}

= 16 (u^{2} + u + \frac{1}{4})

Distribute parentheses

= 16 u^{2} + 16 u + 16 \cdot \frac{1}{4}

16 \cdot \frac{1}{4} = 4

16 \cdot \frac{1}{4}

Multiply fractions:

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

= \frac{1 \cdot 16}{4}

Multiply the numbers:

1 \cdot 16 = 16

= \frac{16}{4}

Divide the numbers:

\frac{16}{4} = 4

= 4

= 16 u^{2} + 16 u + 4

16 u^{2} + 16 u + 4 = 8 u + 13

Move

13

to the left side

16 u^{2} + 16 u + 4 = 8 u + 13

Subtract

13

from both sides

16 u^{2} + 16 u + 4 - 13 = 8 u + 13 - 13

Simplify

16 u^{2} + 16 u - 9 = 8 u

16 u^{2} + 16 u - 9 = 8 u

Move

8 u

to the left side

16 u^{2} + 16 u - 9 = 8 u

Subtract

8 u

from both sides

16 u^{2} + 16 u - 9 - 8 u = 8 u - 8 u

Simplify

16 u^{2} + 8 u - 9 = 0

16 u^{2} + 8 u - 9 = 0

Solve with the quadratic formula

16 u^{2} + 8 u - 9 = 0

Quadratic Equation Formula:

For

a = 16, b = 8, c = - 9

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 16 ( - 9 )}{2 \cdot 16}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 16 ( - 9 )}{2 \cdot 16}

8^{2} - 4 \cdot 16 (- 9) = 810

8^{2} - 4 \cdot 16 (- 9)

Apply rule

- (- a) = a

= 8^{2} + 4 \cdot 16 \cdot 9

Multiply the numbers:

4 \cdot 16 \cdot 9 = 576

= 8^{2} + 576

8^{2} = 64

= 64 + 576

Add the numbers:

64 + 576 = 640

= 640

Prime factorization of

640 : 2^{7} \cdot 5

640

640

divides by

2640 = 320 \cdot 2

= 2 \cdot 320

320

divides by

2320 = 160 \cdot 2

= 2 \cdot 2 \cdot 160

160

divides by

2160 = 80 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 80

80

divides by

280 = 40 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 40

40

divides by

240 = 20 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 20

20

divides by

220 = 10 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 10

10

divides by

210 = 5 \cdot 2

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

2, 5

are all prime numbers, therefore no further factorization is possible

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

= 2^{7} \cdot 5

= 2^{7} \cdot 5

Apply exponent rule:

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

= 2^{6} \cdot 2 \cdot 5

Apply radical rule:

= 2^{6} 2 \cdot 5

Apply radical rule:

2^{6} = 2^{\frac{6}{2}} = 2^{3}

= 2^{3} 2 \cdot 5

Refine

= 810

u_{1, 2} = \frac{- 8 \pm 8 10}{2 \cdot 16}

Separate the solutions

u_{1} = \frac{- 8 + 8 10}{2 \cdot 16}, u_{2} = \frac{- 8 - 8 10}{2 \cdot 16}

u = \frac{- 8 + 8 10}{2 \cdot 16} : \frac{- 1 + 10}{4}

\frac{- 8 + 8 10}{2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{- 8 + 8 10}{32}

Factor

- 8 + 810 : 8 (- 1 + 10)

- 8 + 810

Rewrite as

= - 8 \cdot 1 + 810

Factor out common term

8

= 8 (- 1 + 10)

= \frac{8 ( - 1 + 10 )}{32}

Cancel the common factor:

8

= \frac{- 1 + 10}{4}

u = \frac{- 8 - 8 10}{2 \cdot 16} : - \frac{1 + 10}{4}

\frac{- 8 - 8 10}{2 \cdot 16}

Multiply the numbers:

2 \cdot 16 = 32

= \frac{- 8 - 8 10}{32}

Factor

- 8 - 810 : - 8 (1 + 10)

- 8 - 810

Rewrite as

= - 8 \cdot 1 - 810

Factor out common term

8

= - 8 (1 + 10)

= - \frac{8 ( 1 + 10 )}{32}

Cancel the common factor:

8

= - \frac{1 + 10}{4}

The solutions to the quadratic equation are:

u = \frac{- 1 + 10}{4}, u = - \frac{1 + 10}{4}

Substitute back

u = sin (x)

sin (x) = \frac{- 1 + 10}{4}, sin (x) = - \frac{1 + 10}{4}

sin (x) = \frac{- 1 + 10}{4}, sin (x) = - \frac{1 + 10}{4}

sin (x) = \frac{- 1 + 10}{4} : x = arcsin (\frac{- 1 + 10}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 10}{4}) + 2 πn

sin (x) = \frac{- 1 + 10}{4}

Apply trig inverse properties

sin (x) = \frac{- 1 + 10}{4}

General solutions for

sin (x) = \frac{- 1 + 10}{4}

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

x = arcsin (\frac{- 1 + 10}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 10}{4}) + 2 πn

x = arcsin (\frac{- 1 + 10}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 10}{4}) + 2 πn

sin (x) = - \frac{1 + 10}{4} :

No Solution

sin (x) = - \frac{1 + 10}{4}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = arcsin (\frac{- 1 + 10}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 10}{4}) + 2 πn

Show solutions in decimal form

x = 0.57111 \dots + 2 πn, x = π - 0.57111 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for [sin(x)+1/2 ]^2= 1/2 sin(x)+13/16 ?
The general solution for [sin(x)+1/2 ]^2= 1/2 sin(x)+13/16 is x=0.57111…+2pin,x=pi-0.57111…+2pin