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

The general solution for sin(4x)+cos(4x)=1 is x=(pin)/2 ,x=(pin)/2+pi/8

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sin(4x)+cos(4x)=1

Solution

sin (4 x) + cos (4 x) = 1

Solution

x = \frac{πn}{2}, x = \frac{πn}{2} + \frac{π}{8}

Degrees

x = 0^{\circ} + 9 0^{\circ} n, x = 22. 5^{\circ} + 9 0^{\circ} n

Solution steps

sin (4 x) + cos (4 x) = 1

Rewrite using trig identities

sin (4 x) + cos (4 x)

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

sin (4 x) + cos (4 x)

Rewrite as

= 2 (\frac{1}{2} sin (4 x) + \frac{1}{2} cos (4 x))

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

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

Use the Angle Sum identity:

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

= 2 sin (4 x + \frac{π}{4})

= 2 sin (4 x + \frac{π}{4})

2 sin (4 x + \frac{π}{4}) = 1

Divide both sides by

2

2 sin (4 x + \frac{π}{4}) = 1

Divide both sides by

2

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

Simplify

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

Simplify

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

\frac{2 sin ( 4 x + \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (4 x + \frac{π}{4})

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

sin (4 x + \frac{π}{4}) = \frac{2}{2}

sin (4 x + \frac{π}{4}) = \frac{2}{2}

sin (4 x + \frac{π}{4}) = \frac{2}{2}

General solutions for

sin (4 x + \frac{π}{4}) = \frac{2}{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}

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

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

Solve

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

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

4 x = 2 πn

Divide both sides by

4

4 x = 2 πn

Divide both sides by

4

\frac{4 x}{4} = \frac{2 πn}{4}

Simplify

x = \frac{πn}{2}

x = \frac{πn}{2}

Solve

4 x + \frac{π}{4} = \frac{3 π}{4} + 2 πn : x = \frac{πn}{2} + \frac{π}{8}

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

4 x + \frac{π}{4} - \frac{π}{4} : 4 x

4 x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= 4 x

Simplify

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

\frac{3 π}{4} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{3 π}{4}

Combine the fractions

- \frac{π}{4} + \frac{3 π}{4} : \frac{π}{2}

Apply rule

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

= \frac{- π + 3 π}{4}

Add similar elements:

- π + 3 π = 2 π

= \frac{2 π}{4}

Cancel the common factor:

2

= \frac{π}{2}

= 2 πn + \frac{π}{2}

4 x = 2 πn + \frac{π}{2}

4 x = 2 πn + \frac{π}{2}

4 x = 2 πn + \frac{π}{2}

Divide both sides by

4

4 x = 2 πn + \frac{π}{2}

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{2 πn}{4} + \frac{\frac{π}{2}}{4} : \frac{πn}{2} + \frac{π}{8}

\frac{2 πn}{4} + \frac{\frac{π}{2}}{4}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

\frac{\frac{π}{2}}{4} = \frac{π}{8}

\frac{\frac{π}{2}}{4}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{π}{8}

= \frac{πn}{2} + \frac{π}{8}

x = \frac{πn}{2} + \frac{π}{8}

x = \frac{πn}{2} + \frac{π}{8}

x = \frac{πn}{2} + \frac{π}{8}

x = \frac{πn}{2}, x = \frac{πn}{2} + \frac{π}{8}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(4x)+cos(4x)=1 ?
The general solution for sin(4x)+cos(4x)=1 is x=(pin)/2 ,x=(pin)/2+pi/8