What is the general solution for sin^2(x+4)=1 ?

The general solution for sin^2(x+4)=1 is x= pi/2+2pin-4,x=(3pi)/2+2pin-4

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sin^2(x+4)=1

Solution

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

Solution

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

Degrees

x = - 139.18311 \dots^{\circ} + 36 0^{\circ} n, x = 40.81688 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

sin (x + 4) = u

u^{2} = 1

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

u^{2} = 1

For

x^{2} = f (a)

the solutions are

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

u = 1, u = - 1

1 = 1

1

Apply rule

1 = 1

= 1

- 1 = - 1

- 1

Apply rule

1 = 1

= - 1

u = 1, u = - 1

Substitute back

u = sin (x + 4)

sin (x + 4) = 1, sin (x + 4) = - 1

sin (x + 4) = 1, sin (x + 4) = - 1

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

sin (x + 4) = 1

General solutions for

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

x + 4 = \frac{π}{2} + 2 πn

Solve

x + 4 = \frac{π}{2} + 2 πn : x = \frac{π}{2} + 2 πn - 4

x + 4 = \frac{π}{2} + 2 πn

Move

4

to the right side

x + 4 = \frac{π}{2} + 2 πn

Subtract

4

from both sides

x + 4 - 4 = \frac{π}{2} + 2 πn - 4

Simplify

x = \frac{π}{2} + 2 πn - 4

x = \frac{π}{2} + 2 πn - 4

x = \frac{π}{2} + 2 πn - 4

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

sin (x + 4) = - 1

General solutions for

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

x + 4 = \frac{3 π}{2} + 2 πn

Solve

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

x + 4 = \frac{3 π}{2} + 2 πn

Move

4

to the right side

x + 4 = \frac{3 π}{2} + 2 πn

Subtract

4

from both sides

x + 4 - 4 = \frac{3 π}{2} + 2 πn - 4

Simplify

x = \frac{3 π}{2} + 2 πn - 4

x = \frac{3 π}{2} + 2 πn - 4

x = \frac{3 π}{2} + 2 πn - 4

Combine all the solutions

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x+4)=1 ?
The general solution for sin^2(x+4)=1 is x= pi/2+2pin-4,x=(3pi)/2+2pin-4