What is the general solution for sin(x)+cos(x)= 2/3 ?

The general solution for sin(x)+cos(x)= 2/3 is x=0.49088…+2pin-pi/4 ,x=pi-0.49088…+2pin-pi/4

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sin(x)+cos(x)= 2/3

Solution

sin (x) + cos (x) = \frac{2}{3}

Solution

x = 0.49088 \dots + 2 πn - \frac{π}{4}, x = π - 0.49088 \dots + 2 πn - \frac{π}{4}

Degrees

x = - 16.87449 \dots^{\circ} + 36 0^{\circ} n, x = 106.87449 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (x) + cos (x) = \frac{2}{3}

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}) = \frac{2}{3}

Divide both sides by

2

2 sin (x + \frac{π}{4}) = \frac{2}{3}

Divide both sides by

2

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

Simplify

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

Simplify

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

\frac{2 sin ( x + \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (x + \frac{π}{4})

Simplify

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

\frac{\frac{2}{3}}{2}

Apply the fraction rule:

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

= \frac{2}{3 2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2}{3 \cdot 2 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= \frac{2 ^{1 - \frac{1}{2}}}{3}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{2 ^{\frac{1}{2}}}{3}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{2}{3}

sin (x + \frac{π}{4}) = \frac{2}{3}

sin (x + \frac{π}{4}) = \frac{2}{3}

sin (x + \frac{π}{4}) = \frac{2}{3}

Apply trig inverse properties

sin (x + \frac{π}{4}) = \frac{2}{3}

General solutions for

sin (x + \frac{π}{4}) = \frac{2}{3}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

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

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

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

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

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

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

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

Show solutions in decimal form

x = 0.49088 \dots + 2 πn - \frac{π}{4}, x = π - 0.49088 \dots + 2 πn - \frac{π}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)+cos(x)= 2/3 ?
The general solution for sin(x)+cos(x)= 2/3 is x=0.49088…+2pin-pi/4 ,x=pi-0.49088…+2pin-pi/4