Check the solution to "Solution of trig equation sin⁡x+cos⁡x=2"

agreseza

Answered question

2022-01-16

Solution of trig equation

\sin x + \cos x = \sqrt{2}

maul124uk

Beginner2022-01-17Added 35 answers

If you square both sides, the algebraic identity

{(a + b)}^{2} = a^{2} + 2 a b + b^{2}

gives you

\sin^{2} (x) + 2 \sin (x) \cos (x) + \cos^{2} (x) = 2

now using Pythagoras and the double angle identity for

\sin

gives

1 + \sin (2 x) = 2

\sin (2 x) = 1

Another way is to use the

\tan (\frac{x}{2})

substituition (Weierstrass Substituion) to obtain the polynomial equation

\frac{2 t}{1 + t^{2}} + \frac{1 - t^{2}}{1 + t^{2}} = \sqrt{2}

RizerMix

Expert2022-01-20Added 656 answers

Hint: If

f (x) = a \sin x + b \cos x

then you can write it in a form

f (x) = A \sin (x + φ)

where

A = \pm \sqrt{a^{2} + b^{2}}

and

\tan φ = \frac{b}{a}

In your case a=b=1 so

A = \sqrt{2}

and

\tan φ = 1 \Rightarrow φ = \frac{π}{4}

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