Trigonometric Equations...1

Before continuing on in the polar world, I want to post four trigonometric equations that I recently found online. I don't understand the steps to solve such equations. The following trigonometric equations are not your typical trigonometric equations. When you have time, please provide a step by step solution to each problem. Thank you so much.

Solve for x.

sin x + cos x = 0

 

sin x + cos x = 0
sqrt(2) sin(x + π/4) = 0

x = 1/4 (4 π n - π), n element Z

 

Sorry but I don't understand what you did here? Is there an algebraic way to solve the given equation for x?

 

this way:
sin(x) + cos (x) =0........divide by cos (x)

sin(x)/ cos (x) + cos (x)/cos (x) =0

tan (x) + 1 =0

tan (x) =-1

x=tan^-1 (-1)

x=-π/4

general solution:

x= 3pi/4+pi* n

 

You divided both sides by cos x. Could we also divide both sides by sin x, if needed?

 

yes, you can also divide both sides by sin x and got cot x

 

Thank you.