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Popular >

sin(x)+cos(x)=a

Solution

sin (x) + cos (x) = a

Solution

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

Solution steps

sin (x) + cos (x) = a

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}) = a

Divide both sides by

2

2 sin (x + \frac{π}{4}) = a

Divide both sides by

2

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

Simplify

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

\frac{a}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{a 2}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2 a}{2}

sin (x + \frac{π}{4}) = \frac{2 a}{2}

sin (x + \frac{π}{4}) = \frac{2 a}{2}

sin (x + \frac{π}{4}) = \frac{2 a}{2}

Apply trig inverse properties

sin (x + \frac{π}{4}) = \frac{2 a}{2}

General solutions for

sin (x + \frac{π}{4}) = \frac{2 a}{2}

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

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

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

Solve

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

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

Simplify

arcsin (\frac{2 a}{2}) + 2 πn : arcsin (\frac{a}{2}) + 2 πn

arcsin (\frac{2 a}{2}) + 2 πn

\frac{2 a}{2} = \frac{a}{2}

\frac{2 a}{2}

Apply radical rule:

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{a}{2}

= arcsin (\frac{a}{2}) + 2 πn

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

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

Solve

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

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

Simplify

π + arcsin (- \frac{2 a}{2}) + 2 πn : π + arcsin (- \frac{a}{2}) + 2 πn

π + arcsin (- \frac{2 a}{2}) + 2 πn

\frac{2 a}{2} = \frac{a}{2}

\frac{2 a}{2}

Apply radical rule:

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

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

= \frac{a}{2}

= π + arcsin (- \frac{a}{2}) + 2 πn

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

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

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

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