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cos(x)-sin(2x)=cos(3x)-sin(4x)

Solution

cos (x) - sin (2 x) = cos (3 x) - sin (4 x)

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 2 πn, x = π + 2 πn, x = 0.94247 \dots + 2 πn, x = π - 0.94247 \dots + 2 πn, x = - 0.31415 \dots + 2 πn, x = π + 0.31415 \dots + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 5 4^{\circ} + 36 0^{\circ} n, x = 12 6^{\circ} + 36 0^{\circ} n, x = - 1 8^{\circ} + 36 0^{\circ} n, x = 19 8^{\circ} + 36 0^{\circ} n

Solution steps

cos (x) - sin (2 x) = cos (3 x) - sin (4 x)

Subtract

cos (3 x) - sin (4 x)

from both sides

cos (x) - sin (2 x) - cos (3 x) + sin (4 x) = 0

Rewrite using trig identities

- cos (3 x) + cos (x) - sin (2 x) + sin (4 x)

Use the Sum to Product identity:

sin (s) - sin (t) = 2 sin (\frac{s - t}{2}) cos (\frac{s + t}{2})

= - cos (3 x) + cos (x) + 2 sin (\frac{4 x - 2 x}{2}) cos (\frac{4 x + 2 x}{2})

2 sin (\frac{4 x - 2 x}{2}) cos (\frac{4 x + 2 x}{2}) = 2 sin (x) cos (3 x)

2 sin (\frac{4 x - 2 x}{2}) cos (\frac{4 x + 2 x}{2})

\frac{4 x - 2 x}{2} = x

\frac{4 x - 2 x}{2}

Add similar elements:

4 x - 2 x = 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

= 2 sin (x) cos (\frac{4 x + 2 x}{2})

\frac{4 x + 2 x}{2} = 3 x

\frac{4 x + 2 x}{2}

Add similar elements:

4 x + 2 x = 6 x

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

= 2 sin (x) cos (3 x)

= - cos (3 x) + cos (x) + 2 sin (x) cos (3 x)

cos (3 x) = 4 cos^{3} (x) - 3 cos (x)

cos (3 x)

Rewrite using trig identities

cos (3 x)

Rewrite as

= cos (2 x + x)

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (2 x) cos (x) - sin (2 x) sin (x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x)

Simplify

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

cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x)

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

2 sin (x) cos (x) sin (x)

Apply exponent rule:

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

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

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

Add the numbers:

1 + 1 = 2

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

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

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

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

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

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

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

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

Expand

(2 cos^{2} (x) - 1) cos (x) - 2 (1 - cos^{2} (x)) cos (x) : 4 cos^{3} (x) - 3 cos (x)

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

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

Expand

cos (x) (2 cos^{2} (x) - 1) : 2 cos^{3} (x) - cos (x)

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

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

Simplify

2 cos^{2} (x) cos (x) - 1 \cdot cos (x) : 2 cos^{3} (x) - cos (x)

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

2 cos^{2} (x) cos (x) = 2 cos^{3} (x)

2 cos^{2} (x) cos (x)

Apply exponent rule:

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

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

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

Add the numbers:

2 + 1 = 3

= 2 cos^{3} (x)

1 \cdot cos (x) = cos (x)

1 cos (x)

Multiply:

1 \cdot cos (x) = cos (x)

= cos (x)

= 2 cos^{3} (x) - cos (x)

= 2 cos^{3} (x) - cos (x)

= 2 cos^{3} (x) - cos (x) - 2 (1 - cos^{2} (x)) cos (x)

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Apply minus-plus rules

- (- a) = a

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

Simplify

- 2 \cdot 1 \cdot cos (x) + 2 cos^{2} (x) cos (x) : - 2 cos (x) + 2 cos^{3} (x)

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

2 \cdot 1 \cdot cos (x) = 2 cos (x)

2 \cdot 1 cos (x)

Multiply the numbers:

2 \cdot 1 = 2

= 2 cos (x)

2 cos^{2} (x) cos (x) = 2 cos^{3} (x)

2 cos^{2} (x) cos (x)

Apply exponent rule:

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

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

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

Add the numbers:

2 + 1 = 3

= 2 cos^{3} (x)

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

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

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

Simplify

2 cos^{3} (x) - cos (x) - 2 cos (x) + 2 cos^{3} (x) : 4 cos^{3} (x) - 3 cos (x)

2 cos^{3} (x) - cos (x) - 2 cos (x) + 2 cos^{3} (x)

Group like terms

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

Add similar elements:

2 cos^{3} (x) + 2 cos^{3} (x) = 4 cos^{3} (x)

= 4 cos^{3} (x) - cos (x) - 2 cos (x)

Add similar elements:

- cos (x) - 2 cos (x) = - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= - (4 cos^{3} (x) - 3 cos (x)) + cos (x) + 2 (4 cos^{3} (x) - 3 cos (x)) sin (x)

Simplify

- (4 cos^{3} (x) - 3 cos (x)) + cos (x) + 2 (4 cos^{3} (x) - 3 cos (x)) sin (x) : - 4 cos^{3} (x) + 4 cos (x) + 8 cos^{3} (x) sin (x) - 6 sin (x) cos (x)

- (4 cos^{3} (x) - 3 cos (x)) + cos (x) + 2 (4 cos^{3} (x) - 3 cos (x)) sin (x)

= - (4 cos^{3} (x) - 3 cos (x)) + cos (x) + 2 sin (x) (4 cos^{3} (x) - 3 cos (x))

- (4 cos^{3} (x) - 3 cos (x)) : - 4 cos^{3} (x) + 3 cos (x)

- (4 cos^{3} (x) - 3 cos (x))

Distribute parentheses

= - (4 cos^{3} (x)) - (- 3 cos (x))

Apply minus-plus rules

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

= - 4 cos^{3} (x) + 3 cos (x)

= - 4 cos^{3} (x) + 3 cos (x) + cos (x) + 2 (4 cos^{3} (x) - 3 cos (x)) sin (x)

Expand

2 sin (x) (4 cos^{3} (x) - 3 cos (x)) : 8 cos^{3} (x) sin (x) - 6 sin (x) cos (x)

2 sin (x) (4 cos^{3} (x) - 3 cos (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 2 sin (x), b = 4 cos^{3} (x), c = 3 cos (x)

= 2 sin (x) \cdot 4 cos^{3} (x) - 2 sin (x) \cdot 3 cos (x)

= 2 \cdot 4 cos^{3} (x) sin (x) - 2 \cdot 3 sin (x) cos (x)

Simplify

2 \cdot 4 cos^{3} (x) sin (x) - 2 \cdot 3 sin (x) cos (x) : 8 cos^{3} (x) sin (x) - 6 sin (x) cos (x)

2 \cdot 4 cos^{3} (x) sin (x) - 2 \cdot 3 sin (x) cos (x)

Multiply the numbers:

2 \cdot 4 = 8

= 8 cos^{3} (x) sin (x) - 2 \cdot 3 sin (x) cos (x)

Multiply the numbers:

2 \cdot 3 = 6

= 8 cos^{3} (x) sin (x) - 6 sin (x) cos (x)

= 8 cos^{3} (x) sin (x) - 6 sin (x) cos (x)

= - 4 cos^{3} (x) + 3 cos (x) + cos (x) + 8 cos^{3} (x) sin (x) - 6 sin (x) cos (x)

Add similar elements:

3 cos (x) + cos (x) = 4 cos (x)

= - 4 cos^{3} (x) + 4 cos (x) + 8 cos^{3} (x) sin (x) - 6 sin (x) cos (x)

= - 4 cos^{3} (x) + 4 cos (x) + 8 cos^{3} (x) sin (x) - 6 sin (x) cos (x)

4 cos (x) - 4 cos^{3} (x) - 6 cos (x) sin (x) + 8 cos^{3} (x) sin (x) = 0

Factor

4 cos (x) - 4 cos^{3} (x) - 6 cos (x) sin (x) + 8 cos^{3} (x) sin (x) : 2 cos (x) (2 - 2 cos^{2} (x) - 3 sin (x) + 4 cos^{2} (x) sin (x))

4 cos (x) - 4 cos^{3} (x) - 6 cos (x) sin (x) + 8 cos^{3} (x) sin (x)

Apply exponent rule:

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

sin (x) cos^{3} (x) = cos (x) cos^{2} (x), cos^{3} (x) = cos (x) cos^{2} (x)

= 4 cos (x) - 4 cos (x) cos^{2} (x) - 6 sin (x) cos (x) + 8 cos (x) cos^{2} (x)

Rewrite

8

4 \cdot 2

Rewrite

- 6

3 \cdot 2

= 2 \cdot 2 cos (x) + 2 \cdot 2 cos (x) cos^{2} (x) + 3 \cdot 2 sin (x) cos (x) + 2 \cdot 2 \cdot 2 cos (x) cos^{2} (x)

Factor out common term

2 cos (x)

= 2 cos (x) (2 - 2 cos^{2} (x) - 3 sin (x) + 4 sin (x) cos^{2} (x))

2 cos (x) (2 - 2 cos^{2} (x) - 3 sin (x) + 4 cos^{2} (x) sin (x)) = 0

Solving each part separately

cos (x) = 0 or 2 - 2 cos^{2} (x) - 3 sin (x) + 4 cos^{2} (x) sin (x) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

General solutions for

cos (x) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

2 - 2 cos^{2} (x) - 3 sin (x) + 4 cos^{2} (x) sin (x) = 0 : x = 2 πn, x = π + 2 πn, x = arcsin (\frac{1 + 5}{4}) + 2 πn, x = π - arcsin (\frac{1 + 5}{4}) + 2 πn, x = arcsin (\frac{1 - 5}{4}) + 2 πn, x = π + arcsin (- \frac{1 - 5}{4}) + 2 πn

2 - 2 cos^{2} (x) - 3 sin (x) + 4 cos^{2} (x) sin (x) = 0

Rewrite using trig identities

2 - 2 cos^{2} (x) - 3 sin (x) + 4 cos^{2} (x) sin (x)

Use the Pythagorean identity:

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

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

= 2 - 2 (1 - sin^{2} (x)) - 3 sin (x) + 4 (1 - sin^{2} (x)) sin (x)

Simplify

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

2 - 2 (1 - sin^{2} (x)) - 3 sin (x) + 4 (1 - sin^{2} (x)) sin (x)

= 2 - 2 (1 - sin^{2} (x)) - 3 sin (x) + 4 sin (x) (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)

Apply minus-plus rules

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

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

= 2 - 2 + 2 sin^{2} (x) - 3 sin (x) + 4 (1 - sin^{2} (x)) sin (x)

Expand

4 sin (x) (1 - sin^{2} (x)) : 4 sin (x) - 4 sin^{3} (x)

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

Apply the distributive law:

a (b - c) = ab - a c

a = 4 sin (x), b = 1, c = sin^{2} (x)

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

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

Simplify

4 \cdot 1 \cdot sin (x) - 4 sin^{2} (x) sin (x) : 4 sin (x) - 4 sin^{3} (x)

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

4 \cdot 1 \cdot sin (x) = 4 sin (x)

4 \cdot 1 \cdot sin (x)

Multiply the numbers:

4 \cdot 1 = 4

= 4 sin (x)

4 sin^{2} (x) sin (x) = 4 sin^{3} (x)

4 sin^{2} (x) sin (x)

Apply exponent rule:

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

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

= 4 sin^{2 + 1} (x)

Add the numbers:

2 + 1 = 3

= 4 sin^{3} (x)

= 4 sin (x) - 4 sin^{3} (x)

= 4 sin (x) - 4 sin^{3} (x)

= 2 - 2 + 2 sin^{2} (x) - 3 sin (x) + 4 sin (x) - 4 sin^{3} (x)

Simplify

2 - 2 + 2 sin^{2} (x) - 3 sin (x) + 4 sin (x) - 4 sin^{3} (x) : 2 sin^{2} (x) + sin (x) - 4 sin^{3} (x)

2 - 2 + 2 sin^{2} (x) - 3 sin (x) + 4 sin (x) - 4 sin^{3} (x)

Add similar elements:

- 3 sin (x) + 4 sin (x) = sin (x)

= 2 - 2 + 2 sin^{2} (x) + sin (x) - 4 sin^{3} (x)

2 - 2 = 0

= 2 sin^{2} (x) + sin (x) - 4 sin^{3} (x)

= 2 sin^{2} (x) + sin (x) - 4 sin^{3} (x)

= 2 sin^{2} (x) + sin (x) - 4 sin^{3} (x)

sin (x) + 2 sin^{2} (x) - 4 sin^{3} (x) = 0

Solve by substitution

sin (x) + 2 sin^{2} (x) - 4 sin^{3} (x) = 0

Let:

sin (x) = u

u + 2 u^{2} - 4 u^{3} = 0

u + 2 u^{2} - 4 u^{3} = 0 : u = 0, u = \frac{1 + 5}{4}, u = \frac{1 - 5}{4}

u + 2 u^{2} - 4 u^{3} = 0

Factor

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

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

Apply exponent rule:

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

u^{2} = uu

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

Factor out common term

- u

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

- u (4 u^{2} - 2 u - 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

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

Solve

4 u^{2} - 2 u - 1 = 0 : u = \frac{1 + 5}{4}, u = \frac{1 - 5}{4}

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 4, b = - 2, c = - 1

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

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

(- 2)^{2} - 4 \cdot 4 (- 1) = 25

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

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

Multiply the numbers:

4 \cdot 4 \cdot 1 = 16

= 2^{2} + 16

2^{2} = 4

= 4 + 16

Add the numbers:

4 + 16 = 20

= 20

Prime factorization of

20 : 2^{2} \cdot 5

20

20

divides by

220 = 10 \cdot 2

= 2 \cdot 10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 5

= 2^{2} \cdot 5

= 2^{2} \cdot 5

Apply radical rule:

= 5 2^{2}

Apply radical rule:

2^{2} = 2

= 25

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

Separate the solutions

u_{1} = \frac{- ( - 2 ) + 2 5}{2 \cdot 4}, u_{2} = \frac{- ( - 2 ) - 2 5}{2 \cdot 4}

u = \frac{- ( - 2 ) + 2 5}{2 \cdot 4} : \frac{1 + 5}{4}

\frac{- ( - 2 ) + 2 5}{2 \cdot 4}

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{2 + 2 5}{8}

Factor

2 + 25 : 2 (1 + 5)

2 + 25

Rewrite as

= 2 \cdot 1 + 25

Factor out common term

2

= 2 (1 + 5)

= \frac{2 ( 1 + 5 )}{8}

Cancel the common factor:

2

= \frac{1 + 5}{4}

u = \frac{- ( - 2 ) - 2 5}{2 \cdot 4} : \frac{1 - 5}{4}

\frac{- ( - 2 ) - 2 5}{2 \cdot 4}

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 4 = 8

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

Factor

2 - 25 : 2 (1 - 5)

2 - 25

Rewrite as

= 2 \cdot 1 - 25

Factor out common term

2

= 2 (1 - 5)

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

Cancel the common factor:

2

= \frac{1 - 5}{4}

The solutions to the quadratic equation are:

u = \frac{1 + 5}{4}, u = \frac{1 - 5}{4}

The solutions are

u = 0, u = \frac{1 + 5}{4}, u = \frac{1 - 5}{4}

Substitute back

u = sin (x)

sin (x) = 0, sin (x) = \frac{1 + 5}{4}, sin (x) = \frac{1 - 5}{4}

sin (x) = 0, sin (x) = \frac{1 + 5}{4}, sin (x) = \frac{1 - 5}{4}

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

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

sin (x) = \frac{1 + 5}{4}

Apply trig inverse properties

sin (x) = \frac{1 + 5}{4}

General solutions for

sin (x) = \frac{1 + 5}{4}

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

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

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

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

sin (x) = \frac{1 - 5}{4}

Apply trig inverse properties

sin (x) = \frac{1 - 5}{4}

General solutions for

sin (x) = \frac{1 - 5}{4}

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

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

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

Combine all the solutions

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

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 2 πn, x = π + 2 πn, x = arcsin (\frac{1 + 5}{4}) + 2 πn, x = π - arcsin (\frac{1 + 5}{4}) + 2 πn, x = arcsin (\frac{1 - 5}{4}) + 2 πn, x = π + arcsin (- \frac{1 - 5}{4}) + 2 πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 2 πn, x = π + 2 πn, x = 0.94247 \dots + 2 πn, x = π - 0.94247 \dots + 2 πn, x = - 0.31415 \dots + 2 πn, x = π + 0.31415 \dots + 2 πn

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