How do you solve the equation $1 + sin(x) = cos(x)? 1 + sin(x) = cos(x)$ ?

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Answer 1 · 2015-04-17T04:42:22

Use the identity: $color(blue)(sin x - cos x = (sqrt2)*sin (x - pi/4)$

$f(x) = 1 + sin x - cos x = 0$
$(sqrt2)*sin (x - pi/4) = -1$
$sin (x - pi/4) = -1/sqrt2$
There are 2 arcs that have the same sin value $(-1/(sqrt2))$

$x_1 - pi/4 = pi + pi/4 = (5pi)/4 -> x_1 = (5pi)/4 + pi/4 = (6pi)/4 = (3pi)/2$

$x_2 - pi/4 = 2pi - pi/4 = (7pi)/4 -> x_2 = (7pi)/4 + pi/4 = (8pi)/4 = 2pi.$

Check the 2 answers: $(3pi)/2$ and $2pi$ :

$x = (3pi)/2 -> sin x = -1$

and $cos x = 0 -> f(x) = 1 - 1 + 0 = 0.$ Correct

$x = 2pi -> sin x = 0$ ,

and $cos x = 1 -> f(x) = 1 + 0 - 1 = 0.$ Correct

How do you solve the equation 1 + sin(x) = cos(x)? 1 + sin(x) = cos(x)1 + sin(x) = cos(x)? 1 + sin(x) = cos(x)?