## amorfide 3 years ago I need someone to explain the CAST diagram to me please? I need to know how to find all possible angles between -360 and 360 for sin cosin and tan

1. asnaseer

It is explained quite well in this video: http://www.youtube.com/watch?v=FpjWXKbCC3w

2. amorfide

@asnaseer thank you but what i need is lets say we have sinx=0.5 x=sin^-1(0.5) i need to know how to get all possible angles

3. asnaseer

ok, what you need to do in this specific case is note that the primary solution to:$\sin(x)=0.5$is:$x=30^0=\frac{\pi}{6}$Now, using CAST we also know that:$\sin(x)=\sin(\pi-x)$therefore:$x=\frac{\pi}{6}\text{ or }x=\frac{5\pi}{6}$Next note that the sine curve repeats itself every $$2\pi$$, so the general solution would be:$x=\frac{\pi}{6}+2n\pi\text{ for }n=-\infty,...,-1,0,1,...,\infty$and:$x=\frac{5\pi}{6}+2n\pi\text{ for }n=-\infty,...,-1,0,1,...,\infty$

4. asnaseer

I hope that makes sense?

5. amorfide

it made sense until you brought n into this... what is n exactly?

6. asnaseer

n is just an integer that can take on any value from $$-\infty$$ to $$+\infty$$. e.g. if n=0, we get:$x=\frac{\pi}{6}\text{ or }x=\frac{5\pi}{6}$All n does is add (or subtract) $$2\pi$$ to this answer depending on whether n is positive or negative.

7. asnaseer

that should say "add (or subtract) multiples of $$2\pi$$..."

8. amorfide

but i have never seen a question give me a value of n? so how would i find values for cosin? cos(x)=0.5 x=cos^-1(0.5)

9. asnaseer

If you haven't been introduced to this general form of the answer, then ignore it for now. So the solutions to your first question would just be:$x=\frac{\pi}{6}\text{ or }x=\frac{5\pi}{6}$ Now for the second question, first tel me what you think the primary solution is to:$\cos(x)=0.5$

10. asnaseer

Hang on - you are supposed to find solutions between -360 and +360, correct?

11. amorfide

yes -360 to 360 sorry helping others too

12. amorfide

i know how to convert to radians

13. asnaseer

So first draw these two solutions we found as follows: |dw:1349974770993:dw|

14. asnaseer

So then the negative angles that also satisfy the first equation will be: |dw:1349974926971:dw|

15. asnaseer

Does that make sense?

16. amorfide

the -210 makes sense, the -330 does not

17. asnaseer

-360 + 150 = -210 -360 + 30 = -330

18. asnaseer

|dw:1349975262412:dw|

19. asnaseer

|dw:1349975300819:dw|

20. asnaseer

Does that help?

21. amorfide

okay so sin is fine cosine?

22. asnaseer

Exactly the same principles

23. asnaseer

|dw:1349975392520:dw|

24. amorfide

thank you! you are like my hero i can do most of the complicated things but this was always just too hard :P i understand it better now, thank you!

25. asnaseer

you are more than welcome my friend! I'm glad I was able to explain it to you. :)