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

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

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sin(θ)+cos(θ)=sqrt(2)

Solution

sin (θ) + cos (θ) = 2

Solution

θ = 2 πn + \frac{π}{4}

Degrees

θ = 4 5^{\circ} + 36 0^{\circ} n

Solution steps

sin (θ) + cos (θ) = 2

Rewrite using trig identities

sin (θ) + cos (θ)

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

sin (θ) + cos (θ)

Rewrite as

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

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 (θ) + sin (\frac{π}{4}) cos (θ))

Use the Angle Sum identity:

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

= 2 sin (θ + \frac{π}{4})

= 2 sin (θ + \frac{π}{4})

2 sin (θ + \frac{π}{4}) = 2

Divide both sides by

2

2 sin (θ + \frac{π}{4}) = 2

Divide both sides by

2

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

Simplify

sin (θ + \frac{π}{4}) = 1

sin (θ + \frac{π}{4}) = 1

General solutions for

sin (θ + \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}

θ + \frac{π}{4} = \frac{π}{2} + 2 πn

θ + \frac{π}{4} = \frac{π}{2} + 2 πn

Solve

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

θ + \frac{π}{4} = \frac{π}{2} + 2 πn

Move

\frac{π}{4}

to the right side

θ + \frac{π}{4} = \frac{π}{2} + 2 πn

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

θ + \frac{π}{4} - \frac{π}{4} : θ

θ + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= θ

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}

θ = 2 πn + \frac{π}{4}

θ = 2 πn + \frac{π}{4}

θ = 2 πn + \frac{π}{4}

θ = 2 πn + \frac{π}{4}

Popular Examples

sin^2(x)-cos^2(x)= 1/2

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

sqrt(2)sin^2(x)+cos(x)=0

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

sec(3x)= 2/(sqrt(3)),0<= x<= 2pi

sec (3 x) = \frac{2}{3}, 0 \leq x \leq 2 π

sec(4x)-2=0

sec (4 x) - 2 = 0

2cos(θ)sin(θ)=cos(θ)

2 cos (θ) sin (θ) = cos (θ)

Frequently Asked Questions (FAQ)

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