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sqrt(cos^2(x)+1/2)+sqrt(sin^2(x)+1/2)=2

Solution

cos^{2} (x) + \frac{1}{2} + sin^{2} (x) + \frac{1}{2} = 2

Solution

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

cos^{2} (x) + \frac{1}{2} + sin^{2} (x) + \frac{1}{2} = 2

Subtract

2

from both sides

\frac{2 cos ^{2} ( x ) + 1}{2} + \frac{2 sin ^{2} ( x ) + 1}{2} - 2 = 0

Rewrite using trig identities

- 2 + \frac{1 + 2 cos ^{2} ( x )}{2} + \frac{1 + 2 sin ^{2} ( x )}{2}

Use the Pythagorean identity:

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

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

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

Expand

1 + 2 (1 - sin^{2} (x)) : - 2 sin^{2} (x) + 3

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

Expand

2 (1 - sin^{2} (x)) : 2 - 2 sin^{2} (x)

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

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 1, c = sin^{2} (x)

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

Multiply the numbers:

2 \cdot 1 = 2

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

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

Add the numbers:

1 + 2 = 3

= - 2 sin^{2} (x) + 3

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

- 2 + \frac{1 + 2 sin ^{2} ( x )}{2} + \frac{3 - 2 sin ^{2} ( x )}{2} = 0

Solve by substitution

- 2 + \frac{1 + 2 sin ^{2} ( x )}{2} + \frac{3 - 2 sin ^{2} ( x )}{2} = 0

Let:

sin (x) = u

- 2 + \frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} = 0

- 2 + \frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

- 2 + \frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} = 0

Remove square roots

- 2 + \frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} = 0

Add

2

to both sides

- 2 + \frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} + 2 = 0 + 2

Simplify

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

Subtract

\frac{3 - 2 u ^{2}}{2}

from both sides

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

Simplify

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

Square both sides:

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

- 2 + \frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} = 0

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

Expand

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

Expand

(2 - \frac{3 - 2 u ^{2}}{2})^{2} : 4 - 4 \frac{3 - 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2}

(2 - \frac{3 - 2 u ^{2}}{2})^{2}

Apply Perfect Square Formula:

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

a = 2, b = \frac{3 - 2 u ^{2}}{2}

= 2^{2} - 2 \cdot 2 \frac{3 - 2 u ^{2}}{2} + (\frac{3 - 2 u ^{2}}{2})^{2}

Simplify

2^{2} - 2 \cdot 2 \frac{3 - 2 u ^{2}}{2} + (\frac{3 - 2 u ^{2}}{2})^{2} : 4 - 4 \frac{3 - 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2}

2^{2} - 2 \cdot 2 \frac{3 - 2 u ^{2}}{2} + (\frac{3 - 2 u ^{2}}{2})^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

2 \cdot 2 \frac{3 - 2 u ^{2}}{2} = 4 \frac{3 - 2 u ^{2}}{2}

2 \cdot 2 \frac{3 - 2 u ^{2}}{2}

Multiply the numbers:

2 \cdot 2 = 4

= 4 \frac{3 - 2 u ^{2}}{2}

(\frac{3 - 2 u ^{2}}{2})^{2} = \frac{3 - 2 u ^{2}}{2}

(\frac{3 - 2 u ^{2}}{2})^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= (\frac{3 - 2 u ^{2}}{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

= \frac{3 - 2 u ^{2}}{2}

= 4 - 4 \frac{3 - 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2}

= 4 - 4 \frac{3 - 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2}

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

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

Subtract

\frac{3 - 2 u ^{2}}{2}

from both sides

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

Simplify

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

Subtract

4

from both sides

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

Simplify

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

Square both sides:

4 u^{4} - 20 u^{2} + 25 = 8 (- 2 u^{2} + 3)

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

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

Expand

(\frac{1 + 2 u ^{2}}{2} - \frac{3 - 2 u ^{2}}{2} - 4)^{2} : 4 u^{4} - 20 u^{2} + 25

(\frac{1 + 2 u ^{2}}{2} - \frac{3 - 2 u ^{2}}{2} - 4)^{2}

Combine the fractions

\frac{2 u ^{2} + 1}{2} - \frac{- 2 u ^{2} + 3}{2} : \frac{1 + 2 u ^{2} - ( 3 - 2 u ^{2} )}{2}

Apply rule

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

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

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

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

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

Expand

1 + 2 u^{2} - (3 - 2 u^{2}) : 4 u^{2} - 2

1 + 2 u^{2} - (3 - 2 u^{2})

- (3 - 2 u^{2}) : - 3 + 2 u^{2}

- (3 - 2 u^{2})

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 3 + 2 u^{2}

= 1 + 2 u^{2} - 3 + 2 u^{2}

Simplify

1 + 2 u^{2} - 3 + 2 u^{2} : 4 u^{2} - 2

1 + 2 u^{2} - 3 + 2 u^{2}

Group like terms

= 2 u^{2} + 2 u^{2} + 1 - 3

Add similar elements:

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

= 4 u^{2} + 1 - 3

Add/Subtract the numbers:

1 - 3 = - 2

= 4 u^{2} - 2

= 4 u^{2} - 2

= \frac{4 u ^{2} - 2}{2}

Factor

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

4 u^{2} - 2

Rewrite as

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

Factor out common term

2

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

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

Divide the numbers:

\frac{2}{2} = 1

= 2 u^{2} - 1

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

Subtract the numbers:

- 1 - 4 = - 5

= (2 u^{2} - 5)^{2}

Apply Perfect Square Formula:

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

a = 2 u^{2}, b = 5

= (2 u^{2})^{2} - 2 \cdot 2 u^{2} \cdot 5 + 5^{2}

Simplify

(2 u^{2})^{2} - 2 \cdot 2 u^{2} \cdot 5 + 5^{2} : 4 u^{4} - 20 u^{2} + 25

(2 u^{2})^{2} - 2 \cdot 2 u^{2} \cdot 5 + 5^{2}

(2 u^{2})^{2} = 4 u^{4}

(2 u^{2})^{2}

Apply exponent rule:

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

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

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

Apply exponent rule:

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

= u^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= u^{4}

= 2^{2} u^{4}

2^{2} = 4

= 4 u^{4}

2 \cdot 2 u^{2} \cdot 5 = 20 u^{2}

2 \cdot 2 u^{2} \cdot 5

Multiply the numbers:

2 \cdot 2 \cdot 5 = 20

= 20 u^{2}

5^{2} = 25

5^{2}

5^{2} = 25

= 25

= 4 u^{4} - 20 u^{2} + 25

= 4 u^{4} - 20 u^{2} + 25

Expand

(- 4 \frac{3 - 2 u ^{2}}{2})^{2} : 8 (- 2 u^{2} + 3)

(- 4 \frac{3 - 2 u ^{2}}{2})^{2}

Apply exponent rule:

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

n

is even

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

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

Apply exponent rule:

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

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

(\frac{3 - 2 u ^{2}}{2})^{2} : \frac{3 - 2 u ^{2}}{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= (\frac{3 - 2 u ^{2}}{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

= \frac{3 - 2 u ^{2}}{2}

= 4^{2} \frac{3 - 2 u ^{2}}{2}

4^{2} = 16

= 16 \frac{3 - 2 u ^{2}}{2}

Refine

= 8 (- 2 u^{2} + 3)

4 u^{4} - 20 u^{2} + 25 = 8 (- 2 u^{2} + 3)

4 u^{4} - 20 u^{2} + 25 = 8 (- 2 u^{2} + 3)

4 u^{4} - 20 u^{2} + 25 = 8 (- 2 u^{2} + 3)

Solve

4 u^{4} - 20 u^{2} + 25 = 8 (- 2 u^{2} + 3) : u = \frac{1}{2}, u = - \frac{1}{2}

4 u^{4} - 20 u^{2} + 25 = 8 (- 2 u^{2} + 3)

Expand

8 (- 2 u^{2} + 3) : - 16 u^{2} + 24

8 (- 2 u^{2} + 3)

Apply the distributive law:

a (b + c) = ab + a c

a = 8, b = - 2 u^{2}, c = 3

= 8 (- 2 u^{2}) + 8 \cdot 3

Apply minus-plus rules

+ (- a) = - a

= - 8 \cdot 2 u^{2} + 8 \cdot 3

Simplify

- 8 \cdot 2 u^{2} + 8 \cdot 3 : - 16 u^{2} + 24

- 8 \cdot 2 u^{2} + 8 \cdot 3

Multiply the numbers:

8 \cdot 2 = 16

= - 16 u^{2} + 8 \cdot 3

Multiply the numbers:

8 \cdot 3 = 24

= - 16 u^{2} + 24

= - 16 u^{2} + 24

4 u^{4} - 20 u^{2} + 25 = - 16 u^{2} + 24

Move

24

to the left side

4 u^{4} - 20 u^{2} + 25 = - 16 u^{2} + 24

Subtract

24

from both sides

4 u^{4} - 20 u^{2} + 25 - 24 = - 16 u^{2} + 24 - 24

Simplify

4 u^{4} - 20 u^{2} + 1 = - 16 u^{2}

4 u^{4} - 20 u^{2} + 1 = - 16 u^{2}

Move

16 u^{2}

to the left side

4 u^{4} - 20 u^{2} + 1 = - 16 u^{2}

Add

16 u^{2}

to both sides

4 u^{4} - 20 u^{2} + 1 + 16 u^{2} = - 16 u^{2} + 16 u^{2}

Simplify

4 u^{4} - 4 u^{2} + 1 = 0

4 u^{4} - 4 u^{2} + 1 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

4 v^{2} - 4 v + 1 = 0

Solve

4 v^{2} - 4 v + 1 = 0 : v = \frac{1}{2}

4 v^{2} - 4 v + 1 = 0

Solve with the quadratic formula

4 v^{2} - 4 v + 1 = 0

Quadratic Equation Formula:

For

a = 4, b = - 4, c = 1

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

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

(- 4)^{2} - 4 \cdot 4 \cdot 1 = 0

(- 4)^{2} - 4 \cdot 4 \cdot 1

Apply exponent rule:

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

n

is even

(- 4)^{2} = 4^{2}

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

Multiply the numbers:

4 \cdot 4 \cdot 1 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

Subtract the numbers:

16 - 16 = 0

= 0

v_{1, 2} = \frac{- ( - 4 ) \pm 0}{2 \cdot 4}

v = \frac{- ( - 4 )}{2 \cdot 4}

\frac{- ( - 4 )}{2 \cdot 4} = \frac{1}{2}

\frac{- ( - 4 )}{2 \cdot 4}

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{4}{8}

Cancel the common factor:

4

= \frac{1}{2}

v = \frac{1}{2}

The solution to the quadratic equation is:

v = \frac{1}{2}

v = \frac{1}{2}

Substitute back

v = u^{2},

solve for

u

Solve

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

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

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

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

The solutions are

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

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

Verify Solutions:

u = \frac{1}{2}

True

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

True

Check the solutions by plugging them into

- 2 + \frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} = 0

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

Plug in

u = \frac{1}{2} :

True

- 2 + \frac{1 + 2 ( \frac{1}{2} ) ^{2}}{2} + \frac{3 - 2 ( \frac{1}{2} ) ^{2}}{2} = 0

- 2 + \frac{1 + 2 ( \frac{1}{2} ) ^{2}}{2} + \frac{3 - 2 ( \frac{1}{2} ) ^{2}}{2} = 0

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

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

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

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

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

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

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

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

= (\frac{1}{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

= \frac{1}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= \frac{1 + 1}{2}

Add the numbers:

1 + 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= 1

Apply rule

1 = 1

= 1

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

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

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

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

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

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

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

= (\frac{1}{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

= \frac{1}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= \frac{3 - 1}{2}

Subtract the numbers:

3 - 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= 1

Apply rule

1 = 1

= 1

= - 2 + 1 + 1

Add/Subtract the numbers:

- 2 + 1 + 1 = 0

= 0

0 = 0

True

Plug in

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

True

- 2 + \frac{1 + 2 ( - \frac{1}{2} ) ^{2}}{2} + \frac{3 - 2 ( - \frac{1}{2} ) ^{2}}{2} = 0

- 2 + \frac{1 + 2 ( - \frac{1}{2} ) ^{2}}{2} + \frac{3 - 2 ( - \frac{1}{2} ) ^{2}}{2} = 0

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

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

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

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

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

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

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

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

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

Apply exponent rule:

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

n

is even

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

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

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

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

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

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

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

= (\frac{1}{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

= \frac{1}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= \frac{1 + 1}{2}

Add the numbers:

1 + 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= 1

Apply rule

1 = 1

= 1

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

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

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

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

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

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

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

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

Apply exponent rule:

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

n

is even

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

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

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

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

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

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

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

= (\frac{1}{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

= \frac{1}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= \frac{3 - 1}{2}

Subtract the numbers:

3 - 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= 1

Apply rule

1 = 1

= 1

= - 2 + 1 + 1

Add/Subtract the numbers:

- 2 + 1 + 1 = 0

= 0

0 = 0

True

The solutions are

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

Substitute back

u = sin (x)

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{2}

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{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sin (x) = - \frac{1}{2}

General solutions for

sin (x) = - \frac{1}{2}

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{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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