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

The general solution for sin(x)+cos(x)=3 is No Solution for x\in\mathbb{R}

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sin(x)+cos(x)=3

sin (x) + cos (x) = 3

No Solution for x \in R

Solution steps

sin (x) + cos (x) = 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}) = 3

Divide both sides by

2

2 sin (x + \frac{π}{4}) = 3

Divide both sides by

2

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

Simplify

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

\frac{3}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{3 2}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{3 2}{2}

sin (x + \frac{π}{4}) = \frac{3 2}{2}

sin (x + \frac{π}{4}) = \frac{3 2}{2}

sin (x + \frac{π}{4}) = \frac{3 2}{2}

- 1 \leq sin (x) \leq 1

No Solution for x \in R

sin^{2} (θ) = \frac{3}{4}

2 sin^{2} (x) + 5 cos (x) = 4

- sin (x) + cos (2 x) = 1

tan (θ) = - 3

cot (θ) = \frac{3}{3}

What is the general solution for sin(x)+cos(x)=3 ?
The general solution for sin(x)+cos(x)=3 is No Solution for x\in\mathbb{R}