What is the general solution for sin(x)+cos(x)=sqrt(2) ?

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

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sin(x)+cos(x)=sqrt(2)

Solution

sin (x) + cos (x) = 2

Solution

x = 2 πn + \frac{π}{4}

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n

Solution steps

sin (x) + cos (x) = 2

Rewrite using trig identities

sin (x) + cos (x)

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

sin (x) + cos (x)

Rewrite as

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

Use the Angle Sum identity:

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

= 2 sin (x + \frac{π}{4})

= 2 sin (x + \frac{π}{4})

2 sin (x + \frac{π}{4}) = 2

Divide both sides by

2

2 sin (x + \frac{π}{4}) = 2

Divide both sides by

2

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

Simplify

sin (x + \frac{π}{4}) = 1

sin (x + \frac{π}{4}) = 1

General solutions for

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

x + \frac{π}{4} = \frac{π}{2} + 2 πn

Solve

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

x + \frac{π}{4} = \frac{π}{2} + 2 πn

Move

\frac{π}{4}

to the right side

x + \frac{π}{4} = \frac{π}{2} + 2 πn

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= x

Simplify

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

\frac{π}{2} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{2} - \frac{π}{4}

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

4

For

\frac{π}{2} :

multiply the denominator and numerator by

2

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

= \frac{π 2}{4} - \frac{π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - π}{4}

Add similar elements:

2 π - π = π

= 2 πn + \frac{π}{4}

x = 2 πn + \frac{π}{4}

x = 2 πn + \frac{π}{4}

x = 2 πn + \frac{π}{4}

x = 2 πn + \frac{π}{4}

Popular Examples

sin(-(3pi)/4)

sin (- \frac{3 π}{4})

4cos^2(θ)-3=0

4 cos^{2} (θ) - 3 = 0

tan(pi/6)

tan (\frac{π}{6})

2sin(2x)=sqrt(3)

2 sin (2 x) = 3

tan((3pi)/4)

tan (\frac{3 π}{4})

Frequently Asked Questions (FAQ)

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