Question

Find all solutions in the interval [0, 2π).

(sin x)(cos x) = 0

Answer 1

I get sin x = 0 or cos x = 0 but how do I get the solutions from that?

Answer 2

do you have a unit circle with you?

Answer 3

Yes

Answer 4

ok to solve sin(x) = 0 for x, you need to find points on the unit circle where the y coordinate is 0

Answer 5

this is because on the unit circle, any point is of the form (x,y) where

x = cos(theta)
y = sin(theta)

theta = angle

Answer 6

That would be pi and 2pi

Answer 7

For cos that would be pi/2 and 3pi/2

Answer 8

0 is coterminal to 2pi

Answer 9

so the solutions to sin(x) = 0 are x = 0 and x = pi

Answer 10

this is on the interval [0, 2pi)

Answer 11

and good, the solutions to cos(x) = 0 are x = pi/2 and x = 3pi/2

Answer 12

so in total, the four solutions are


x = 0, x = pi, x = pi/2, x = 3pi/2

Answer 13

\[\frac{ 1 }{2} (2\sin x \cos x)=0,\sin 2x=0=\sin n \pi \]
2x=n pi
\[x=\frac{ n \pi }{ 2 }\]
for n=0,1,2,3
calculate x

Answer 14

Got it, thanks

Answer 15

yw