What is the general solution for 4sin^2(x-60)=3 ?

The general solution for 4sin^2(x-60)=3 is x=120+360n,x=360n+180,x=360n+300,x=360n+360

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4sin^2(x-60)=3

Solution

4 sin^{2} (x - 6 0^{\circ}) = 3

Solution

x = 12 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} n + 18 0^{\circ}, x = 36 0^{\circ} n + 30 0^{\circ}, x = 36 0^{\circ} n + 36 0^{\circ}

Radians

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

Solution steps

4 sin^{2} (x - 6 0^{\circ}) = 3

Solve by substitution

4 sin^{2} (x - 6 0^{\circ}) = 3

Let:

sin (x - 6 0^{\circ}) = u

4 u^{2} = 3

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

4 u^{2} = 3

Divide both sides by

4

4 u^{2} = 3

Divide both sides by

4

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

Simplify

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

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

For

x^{2} = f (a)

the solutions are

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

u = \frac{3}{4}, u = - \frac{3}{4}

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

\frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

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

- \frac{3}{4}

Simplify

\frac{3}{4} : \frac{3}{2}

\frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

= - \frac{3}{2}

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

Substitute back

u = sin (x - 6 0^{\circ})

sin (x - 6 0^{\circ}) = \frac{3}{2}, sin (x - 6 0^{\circ}) = - \frac{3}{2}

sin (x - 6 0^{\circ}) = \frac{3}{2}, sin (x - 6 0^{\circ}) = - \frac{3}{2}

sin (x - 6 0^{\circ}) = \frac{3}{2} : x = 12 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} n + 18 0^{\circ}

sin (x - 6 0^{\circ}) = \frac{3}{2}

General solutions for

sin (x - 6 0^{\circ}) = \frac{3}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x - 6 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n, x - 6 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n

x - 6 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n, x - 6 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n

Solve

x - 6 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n : x = 12 0^{\circ} + 36 0^{\circ} n

x - 6 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n

Move

6 0^{\circ}

to the right side

x - 6 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n

Add

6 0^{\circ}

to both sides

x - 6 0^{\circ} + 6 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplify

x - 6 0^{\circ} + 6 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplify

x - 6 0^{\circ} + 6 0^{\circ} : x

x - 6 0^{\circ} + 6 0^{\circ}

Add similar elements:

- 6 0^{\circ} + 6 0^{\circ} = 0

= x

Simplify

6 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ} : 12 0^{\circ} + 36 0^{\circ} n

6 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Group like terms

= 6 0^{\circ} + 6 0^{\circ} + 36 0^{\circ} n

Combine the fractions

6 0^{\circ} + 6 0^{\circ} : 12 0^{\circ}

Apply rule

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

= \frac{18 0 ^{\circ} + 18 0 ^{\circ}}{3}

Add similar elements:

18 0^{\circ} + 18 0^{\circ} = 36 0^{\circ}

= 12 0^{\circ}

= 12 0^{\circ} + 36 0^{\circ} n

x = 12 0^{\circ} + 36 0^{\circ} n

x = 12 0^{\circ} + 36 0^{\circ} n

x = 12 0^{\circ} + 36 0^{\circ} n

Solve

x - 6 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 18 0^{\circ}

x - 6 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n

Move

6 0^{\circ}

to the right side

x - 6 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n

Add

6 0^{\circ}

to both sides

x - 6 0^{\circ} + 6 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplify

x - 6 0^{\circ} + 6 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplify

x - 6 0^{\circ} + 6 0^{\circ} : x

x - 6 0^{\circ} + 6 0^{\circ}

Add similar elements:

- 6 0^{\circ} + 6 0^{\circ} = 0

= x

Simplify

12 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ} : 36 0^{\circ} n + 18 0^{\circ}

12 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Group like terms

= 36 0^{\circ} n + 6 0^{\circ} + 12 0^{\circ}

Combine the fractions

6 0^{\circ} + 12 0^{\circ} : 18 0^{\circ}

Apply rule

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

= \frac{18 0 ^{\circ} + 36 0 ^{\circ}}{3}

Add similar elements:

18 0^{\circ} + 36 0^{\circ} = 54 0^{\circ}

= 18 0^{\circ}

Divide the numbers:

\frac{3}{3} = 1

= 18 0^{\circ}

= 36 0^{\circ} n + 18 0^{\circ}

x = 36 0^{\circ} n + 18 0^{\circ}

x = 36 0^{\circ} n + 18 0^{\circ}

x = 36 0^{\circ} n + 18 0^{\circ}

x = 12 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} n + 18 0^{\circ}

sin (x - 6 0^{\circ}) = - \frac{3}{2} : x = 36 0^{\circ} n + 30 0^{\circ}, x = 36 0^{\circ} n + 36 0^{\circ}

sin (x - 6 0^{\circ}) = - \frac{3}{2}

General solutions for

sin (x - 6 0^{\circ}) = - \frac{3}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x - 6 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n, x - 6 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

x - 6 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n, x - 6 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

Solve

x - 6 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 30 0^{\circ}

x - 6 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n

Move

6 0^{\circ}

to the right side

x - 6 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n

Add

6 0^{\circ}

to both sides

x - 6 0^{\circ} + 6 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplify

x - 6 0^{\circ} + 6 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplify

x - 6 0^{\circ} + 6 0^{\circ} : x

x - 6 0^{\circ} + 6 0^{\circ}

Add similar elements:

- 6 0^{\circ} + 6 0^{\circ} = 0

= x

Simplify

24 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ} : 36 0^{\circ} n + 30 0^{\circ}

24 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Group like terms

= 36 0^{\circ} n + 6 0^{\circ} + 24 0^{\circ}

Combine the fractions

6 0^{\circ} + 24 0^{\circ} : 30 0^{\circ}

Apply rule

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

= \frac{18 0 ^{\circ} + 72 0 ^{\circ}}{3}

Add similar elements:

18 0^{\circ} + 72 0^{\circ} = 90 0^{\circ}

= 30 0^{\circ}

= 36 0^{\circ} n + 30 0^{\circ}

x = 36 0^{\circ} n + 30 0^{\circ}

x = 36 0^{\circ} n + 30 0^{\circ}

x = 36 0^{\circ} n + 30 0^{\circ}

Solve

x - 6 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 36 0^{\circ}

x - 6 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

Move

6 0^{\circ}

to the right side

x - 6 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

Add

6 0^{\circ}

to both sides

x - 6 0^{\circ} + 6 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplify

x - 6 0^{\circ} + 6 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplify

x - 6 0^{\circ} + 6 0^{\circ} : x

x - 6 0^{\circ} + 6 0^{\circ}

Add similar elements:

- 6 0^{\circ} + 6 0^{\circ} = 0

= x

Simplify

30 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ} : 36 0^{\circ} n + 36 0^{\circ}

30 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Group like terms

= 36 0^{\circ} n + 6 0^{\circ} + 30 0^{\circ}

Combine the fractions

6 0^{\circ} + 30 0^{\circ} : 36 0^{\circ}

Apply rule

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

= \frac{18 0 ^{\circ} + 90 0 ^{\circ}}{3}

Add similar elements:

18 0^{\circ} + 90 0^{\circ} = 108 0^{\circ}

= 36 0^{\circ}

Divide the numbers:

\frac{6}{3} = 2

= 36 0^{\circ}

= 36 0^{\circ} n + 36 0^{\circ}

x = 36 0^{\circ} n + 36 0^{\circ}

x = 36 0^{\circ} n + 36 0^{\circ}

x = 36 0^{\circ} n + 36 0^{\circ}

x = 36 0^{\circ} n + 30 0^{\circ}, x = 36 0^{\circ} n + 36 0^{\circ}

Combine all the solutions

x = 12 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} n + 18 0^{\circ}, x = 36 0^{\circ} n + 30 0^{\circ}, x = 36 0^{\circ} n + 36 0^{\circ}

Popular Examples

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Frequently Asked Questions (FAQ)

What is the general solution for 4sin^2(x-60)=3 ?
The general solution for 4sin^2(x-60)=3 is x=120+360n,x=360n+180,x=360n+300,x=360n+360