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

1 Answer
Apr 17, 2015

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