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sin(x)+1=2sqrt(1-sin^2(x))

Solution

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

Solution

x = 0.64350 \dots + 2 πn, x = π - 0.64350 \dots + 2 πn, x = \frac{3 π}{2} + 2 πn

Degrees

x = 36.86989 \dots^{\circ} + 36 0^{\circ} n, x = 143.13010 \dots^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

sin (x) = u

u + 1 = 2 1 - u^{2}

u + 1 = 2 1 - u^{2} : u = \frac{3}{5}, u = - 1

u + 1 = 2 1 - u^{2}

Square both sides:

u^{2} + 2 u + 1 = 4 - 4 u^{2}

u + 1 = 2 1 - u^{2}

(u + 1)^{2} = (2 1 - u^{2})^{2}

Expand

(u + 1)^{2} : u^{2} + 2 u + 1

(u + 1)^{2}

Apply Perfect Square Formula:

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

a = u, b = 1

= u^{2} + 2 u \cdot 1 + 1^{2}

Simplify

u^{2} + 2 u \cdot 1 + 1^{2} : u^{2} + 2 u + 1

u^{2} + 2 u \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= u^{2} + 2 \cdot 1 \cdot u + 1

Multiply the numbers:

2 \cdot 1 = 2

= u^{2} + 2 u + 1

= u^{2} + 2 u + 1

Expand

(2 1 - u^{2})^{2} : 4 - 4 u^{2}

(2 1 - u^{2})^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (1 - u^{2})^{2}

(1 - u^{2})^{2} : 1 - u^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= (1 - u^{2})^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 1 - u^{2}

= 2^{2} (1 - u^{2})

2^{2} = 4

= 4 (1 - u^{2})

Expand

4 (1 - u^{2}) : 4 - 4 u^{2}

4 (1 - u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = u^{2}

= 4 \cdot 1 - 4 u^{2}

Multiply the numbers:

4 \cdot 1 = 4

= 4 - 4 u^{2}

= 4 - 4 u^{2}

u^{2} + 2 u + 1 = 4 - 4 u^{2}

u^{2} + 2 u + 1 = 4 - 4 u^{2}

Solve

u^{2} + 2 u + 1 = 4 - 4 u^{2} : u = \frac{3}{5}, u = - 1

u^{2} + 2 u + 1 = 4 - 4 u^{2}

Move

4 u^{2}

to the left side

u^{2} + 2 u + 1 = 4 - 4 u^{2}

Add

4 u^{2}

to both sides

u^{2} + 2 u + 1 + 4 u^{2} = 4 - 4 u^{2} + 4 u^{2}

Simplify

5 u^{2} + 2 u + 1 = 4

5 u^{2} + 2 u + 1 = 4

Move

4

to the left side

5 u^{2} + 2 u + 1 = 4

Subtract

4

from both sides

5 u^{2} + 2 u + 1 - 4 = 4 - 4

Simplify

5 u^{2} + 2 u - 3 = 0

5 u^{2} + 2 u - 3 = 0

Solve with the quadratic formula

5 u^{2} + 2 u - 3 = 0

Quadratic Equation Formula:

For

a = 5, b = 2, c = - 3

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 5 ( - 3 )}{2 \cdot 5}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 5 ( - 3 )}{2 \cdot 5}

2^{2} - 4 \cdot 5 (- 3) = 8

2^{2} - 4 \cdot 5 (- 3)

Apply rule

- (- a) = a

= 2^{2} + 4 \cdot 5 \cdot 3

Multiply the numbers:

4 \cdot 5 \cdot 3 = 60

= 2^{2} + 60

2^{2} = 4

= 4 + 60

Add the numbers:

4 + 60 = 64

= 64

Factor the number:

64 = 8^{2}

= 8^{2}

Apply radical rule:

8^{2} = 8

= 8

u_{1, 2} = \frac{- 2 \pm 8}{2 \cdot 5}

Separate the solutions

u_{1} = \frac{- 2 + 8}{2 \cdot 5}, u_{2} = \frac{- 2 - 8}{2 \cdot 5}

u = \frac{- 2 + 8}{2 \cdot 5} : \frac{3}{5}

\frac{- 2 + 8}{2 \cdot 5}

Add/Subtract the numbers:

- 2 + 8 = 6

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

Multiply the numbers:

2 \cdot 5 = 10

= \frac{6}{10}

Cancel the common factor:

2

= \frac{3}{5}

u = \frac{- 2 - 8}{2 \cdot 5} : - 1

\frac{- 2 - 8}{2 \cdot 5}

Subtract the numbers:

- 2 - 8 = - 10

= \frac{- 10}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 10}{10}

Apply the fraction rule:

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

= - \frac{10}{10}

Apply rule

\frac{a}{a} = 1

= - 1

The solutions to the quadratic equation are:

u = \frac{3}{5}, u = - 1

u = \frac{3}{5}, u = - 1

Verify Solutions:

u = \frac{3}{5}

True

, u = - 1

True

Check the solutions by plugging them into

u + 1 = 2 1 - u^{2}

Remove the ones that don't agree with the equation.

Plug in

u = \frac{3}{5} :

True

(\frac{3}{5}) + 1 = 2 1 - (\frac{3}{5})^{2}

(\frac{3}{5}) + 1 = \frac{8}{5}

(\frac{3}{5}) + 1

Remove parentheses:

(a) = a

= \frac{3}{5} + 1

Convert element to fraction:

1 = \frac{1 \cdot 5}{5}

= \frac{1 \cdot 5}{5} + \frac{3}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 5 + 3}{5}

1 \cdot 5 + 3 = 8

1 \cdot 5 + 3

Multiply the numbers:

1 \cdot 5 = 5

= 5 + 3

Add the numbers:

5 + 3 = 8

= 8

= \frac{8}{5}

2 1 - (\frac{3}{5})^{2} = \frac{8}{5}

2 1 - (\frac{3}{5})^{2}

1 - (\frac{3}{5})^{2} = \frac{4}{5}

1 - (\frac{3}{5})^{2}

(\frac{3}{5})^{2} = \frac{9}{25}

(\frac{3}{5})^{2}

Apply exponent rule:

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

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

3^{2} = 9

= \frac{9}{5 ^{2}}

5^{2} = 25

= \frac{9}{25}

= 1 - \frac{9}{25}

Join

1 - \frac{9}{25} : \frac{16}{25}

1 - \frac{9}{25}

Convert element to fraction:

1 = \frac{1 \cdot 25}{25}

= \frac{1 \cdot 25}{25} - \frac{9}{25}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 25 - 9}{25}

1 \cdot 25 - 9 = 16

1 \cdot 25 - 9

Multiply the numbers:

1 \cdot 25 = 25

= 25 - 9

Subtract the numbers:

25 - 9 = 16

= 16

= \frac{16}{25}

= \frac{16}{25}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{16}{25}

25 = 5

25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

= \frac{16}{5}

16 = 4

16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

= \frac{4}{5}

= 2 \cdot \frac{4}{5}

Multiply fractions:

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

= \frac{4 \cdot 2}{5}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{8}{5}

\frac{8}{5} = \frac{8}{5}

True

Plug in

u = - 1 :

True

(- 1) + 1 = 2 1 - (- 1)^{2}

(- 1) + 1 = 0

(- 1) + 1

Remove parentheses:

(- a) = - a

= - 1 + 1

Add/Subtract the numbers:

- 1 + 1 = 0

= 0

2 1 - (- 1)^{2} = 0

2 1 - (- 1)^{2}

1 - (- 1)^{2} = 0

1 - (- 1)^{2}

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

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

n

is even

(- 1)^{2} = 1^{2}

= 1^{2}

Apply rule

1^{a} = 1

= 1

= 1 - 1

Subtract the numbers:

1 - 1 = 0

= 0

Apply rule

0 = 0

= 0

= 2 \cdot 0

Apply rule

0 \cdot a = 0

= 0

0 = 0

True

The solutions are

u = \frac{3}{5}, u = - 1

Substitute back

u = sin (x)

sin (x) = \frac{3}{5}, sin (x) = - 1

sin (x) = \frac{3}{5}, sin (x) = - 1

sin (x) = \frac{3}{5} : x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

sin (x) = \frac{3}{5}

Apply trig inverse properties

sin (x) = \frac{3}{5}

General solutions for

sin (x) = \frac{3}{5}

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

x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

General solutions for

sin (x) = - 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}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn, x = \frac{3 π}{2} + 2 πn

Show solutions in decimal form

x = 0.64350 \dots + 2 πn, x = π - 0.64350 \dots + 2 πn, x = \frac{3 π}{2} + 2 πn

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