Reference angles question (confusion on my part)

Jul 2015
617
16
United States
Let us say find cosine 230 do I always subtract it from 270? Or would I do 230-180. It looks like on my notes I did it sometimes from both ways which way is the correct way? For instance I had a problem where I had sin being 15.5 but it was also 165.5 why wasn't it just 105.5??? Because on my other problem. Sin 45 degrees can also be sin 135 and do you not get that from adding 45+90??

Does anyone see what I'm trying to say here or am I not being clear >.< , I've been wondering about this for a while now
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Better to use 180 because is it less than 230. Basically, you need an angle between 0 and 90 degrees. Using 230- 180= 50 you would argue that sin(50)= 0.7660 and cos(50)= 0.6428 but since 230 degrees is in the third quadrant, where both sine and cosine are negative, sin(230)= -0.7660 and cos(230)= -0.6428. On the other hand, 230- 270= -40. That is NOT positive so not between 0 and 90 degrees. It could be used if you use sin(-40)= -sin(40) and cos(-40)= cos(40).
 

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
Reference angles are always the angle between the terminal side of $\theta$ and the nearest side of the x-axis.

If you make a sketch, then determining the reference angle becomes easy to see ...

$230^\circ = 180^\circ + 50^\circ$
 

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