# Math Help - Simple Trigonometric Equations

1. ## Simple Trigonometric Equations

These are hard for me right now, it will be fine once I understand it but I'm not sure because it's very confusing!

If I had cos 3x = 0.5, would I:
Use cos^-1 0.5 = 60
and because cos is positive in the second quadrant, would I add the 60 to 90 to make 150...
then divide both answers by 3 to get 20 and 50?

Is this right, and would it work for any other including sin?

Also if I had tan x = -root 3, would I solve it like:
tan^-1 root 3 = 60
and because tan is positive in the third quadrant, would I add the 60 again to 180 to make 240
and are my solutions 60 and 240, and nothing else?

2. ## Trigonometry

Hello db5vry
Originally Posted by db5vry
If I had cos 3x = 0.5, would I:
Use cos^-1 0.5 = 60
and because cos is positive in the second quadrant, would I add the 60 to 90 to make 150...
then divide both answers by 3 to get 20 and 50?
Look at the attached diagram, showing $\cos 60^o = 0.5$. The other angle in the diagram whose cosine is also $0.5$ is in the fourth quadrant: it is $360^o - 60^o = 300^o$.

If we allow angles of more than $360^o$, the next angle whose cosine is 0.5 is $360 + 60 = 420^o$; the next is $660^o$, and so on.

So if $\cos 3x = 0.5$, the possible values of $3x$ are $60^o, 300^o, 420^o, 660^o, \dots$

To find the values of $x$, then, you'll need to divide each of these by $3$ to get:

$x = 20^o, 100^o, 140^o, 220^o, \dots$

(Cosine is positive in the first and fourth quadrants, of course.)

Hello db5vryLook at the attached diagram, showing $\cos 60^o = 0.5$. The other angle in the diagram whose cosine is also $0.5$ is in the fourth quadrant: it is $360^o - 60^o = 300^o$.

If we allow angles of more than $360^o$, the next angle whose cosine is 0.5 is $360 + 60 = 420^o$; the next is $660^o$, and so on.

So if $\cos 3x = 0.5$, the possible values of $3x$ are $60^o, 300^o, 420^o, 660^o, \dots$

To find the values of $x$, then, you'll need to divide each of these by $3$ to get:

$x = 20^o, 100^o, 140^o, 220^o, \dots$

(Cosine is positive in the first and fourth quadrants, of course.)