What is the general solution for sin(x)+cos(x)=0.2 ?

The general solution for sin(x)+cos(x)=0.2 is x=0.14189…+2pin-pi/4 ,x=pi-0.14189…+2pin-pi/4

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sin(x)+cos(x)=0.2

Solution

sin (x) + cos (x) = 0.2

Solution

x = 0.14189 \dots + 2 πn - \frac{π}{4}, x = π - 0.14189 \dots + 2 πn - \frac{π}{4}

Degrees

x = - 36.86989 \dots^{\circ} + 36 0^{\circ} n, x = 126.86989 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (x) + cos (x) = 0.2

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}) = 0.2

Divide both sides by

2

2 sin (x + \frac{π}{4}) = 0.2

Divide both sides by

2

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{0.2}{2}

Simplify

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{0.2}{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{0.2}{2} : 0.14142 \dots

\frac{0.2}{2}

Convert element to a decimal form

2 = 1.41421 \dots

= \frac{0.2}{1.41421 \dots}

Divide the numbers:

\frac{0.2}{1.41421 \dots} = 0.14142 \dots

= 0.14142 \dots

sin (x + \frac{π}{4}) = 0.14142 \dots

sin (x + \frac{π}{4}) = 0.14142 \dots

sin (x + \frac{π}{4}) = 0.14142 \dots

Apply trig inverse properties

sin (x + \frac{π}{4}) = 0.14142 \dots

General solutions for

sin (x + \frac{π}{4}) = 0.14142 \dots

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

x + \frac{π}{4} = arcsin (0.14142 \dots) + 2 πn, x + \frac{π}{4} = π - arcsin (0.14142 \dots) + 2 πn

x + \frac{π}{4} = arcsin (0.14142 \dots) + 2 πn, x + \frac{π}{4} = π - arcsin (0.14142 \dots) + 2 πn

Solve

x + \frac{π}{4} = arcsin (0.14142 \dots) + 2 πn : x = arcsin (0.14142 \dots) + 2 πn - \frac{π}{4}

x + \frac{π}{4} = arcsin (0.14142 \dots) + 2 πn

Move

\frac{π}{4}

to the right side

x + \frac{π}{4} = arcsin (0.14142 \dots) + 2 πn

Subtract

\frac{π}{4}

from both sides

x + \frac{π}{4} - \frac{π}{4} = arcsin (0.14142 \dots) + 2 πn - \frac{π}{4}

Simplify

x = arcsin (0.14142 \dots) + 2 πn - \frac{π}{4}

x = arcsin (0.14142 \dots) + 2 πn - \frac{π}{4}

Solve

x + \frac{π}{4} = π - arcsin (0.14142 \dots) + 2 πn : x = π - arcsin (0.14142 \dots) + 2 πn - \frac{π}{4}

x + \frac{π}{4} = π - arcsin (0.14142 \dots) + 2 πn

Move

\frac{π}{4}

to the right side

x + \frac{π}{4} = π - arcsin (0.14142 \dots) + 2 πn

Subtract

\frac{π}{4}

from both sides

x + \frac{π}{4} - \frac{π}{4} = π - arcsin (0.14142 \dots) + 2 πn - \frac{π}{4}

Simplify

x = π - arcsin (0.14142 \dots) + 2 πn - \frac{π}{4}

x = π - arcsin (0.14142 \dots) + 2 πn - \frac{π}{4}

x = arcsin (0.14142 \dots) + 2 πn - \frac{π}{4}, x = π - arcsin (0.14142 \dots) + 2 πn - \frac{π}{4}

Show solutions in decimal form

x = 0.14189 \dots + 2 πn - \frac{π}{4}, x = π - 0.14189 \dots + 2 πn - \frac{π}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)+cos(x)=0.2 ?
The general solution for sin(x)+cos(x)=0.2 is x=0.14189…+2pin-pi/4 ,x=pi-0.14189…+2pin-pi/4