What is the general solution for sin(x)+cos(x)=4 ?

The general solution for sin(x)+cos(x)=4 is No Solution for x\in\mathbb{R}

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sin(x)+cos(x)=4

sin (x) + cos (x) = 4

No Solution for x \in R

Solution steps

sin (x) + cos (x) = 4

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}) = 4

Divide both sides by

2

2 sin (x + \frac{π}{4}) = 4

Divide both sides by

2

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

Simplify

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{4}{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{4}{2} : 22

\frac{4}{2}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

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

Cancel

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

\frac{2 ^{2}}{2}

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

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

2^{\frac{3}{2}} = 22

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

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

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 22

= 22

sin (x + \frac{π}{4}) = 22

sin (x + \frac{π}{4}) = 22

sin (x + \frac{π}{4}) = 22

- 1 \leq sin (x) \leq 1

No Solution for x \in R

x, ln (y) = sin (x)

sin (θ) + cos (θ) = \frac{3}{2}

sin (x) + \frac{π}{2} = cos (x)

5 cos (x) - 3 = 9 cos (x) - 5

3 sin^{2} (x) - 8 sin (x) + 4 = 0

What is the general solution for sin(x)+cos(x)=4 ?
The general solution for sin(x)+cos(x)=4 is No Solution for x\in\mathbb{R}