Solving Trigonometric Equations...6

Section 5.3

 

38.

sec(x)+tan(x)=1 .........interval [0,2pi]

1/cos(x)+sin(x)/cos(x)=1

(1+sin(x))/cos(x)=1

1+sin(x)-cos(x)=0

rewrite using identity sin(x)-cos(x)=sqrt(2)sin(x-pi/4)

1+sqrt(2)sin(x-pi/4)=0

sqrt(2)sin(x-pi/4)=-1

sin(x-pi/4)=-1/sqrt(2)

sin(x-pi/4)=-sqrt(2)/2

general solutions:

Radians:
x-pi/4=5pi/4+2pi*k
x-pi/4=7pi/4+2pi*k
solve for x
x=5pi/4+2pi*k+pi/4 =>x=3pi/2+2pi*k
x=7pi/4+2pi*k+pi/4=>x=2pi++2pi*k

solutions for given interval:
x=3pi/2+2pi*k
x=2pi++2pi*k
Degrees:
x=0°
x=360°

 

I didn't know the following trigonometric identity:

sin(x)-cos(x)=sqrt(2)sin(x-pi/4)

This is why I got stuck. I will do a few more today.