I was into a trig book for a while and something clicked me which I do not understand so I am asking it here.
Suppose I need to find the value $Sin 120^\circ$, I know then, I have to take a reference angle (or triangle) in quadrant 2 as shown below.
But, how would the reference angle of $Sin210^\circ$ look like? In which quadrant would it fall? I have listed 2 approaches below. Help me track down the correct one.
Approach 1 This?
Or Approach 2?
Look, in $Sin 120^\circ$, we subtract $60^\circ$ from $180^\circ$ (as $180^\circ$ is the bigger angle here) and proceed on to find out the values of $Sin 60^\circ$ as it is the reference angle there.
Then, Shouldn't Approach 2 be correct as there I am subtracting $60^\circ$ from $270^\circ$ (as $270^\circ$ is the bigger angle here) and make the reference angle on $Y-Axis$?
Also please, be kind enough to enlighten me with the reasons as to why one of the 2 approaches is correct.
Thank You. ;)
$\endgroup$1 Answer
$\begingroup$Your approach number $1$ is correct. The reference angle should always be between your ray of interest and the $x$-axis. The reference angle for $210^\circ$ is $210^\circ-180^\circ=30^\circ$. This is because the cosine and sine of $210^\circ$ are the same as the cosine and sine of $30^\circ$, except for both of them being negative.
Reference angles aren't relative to an angle that's necessarily bigger or smaller in every case. In the first quadrant, the reference angle of $\theta$ is $\theta-0^\circ$. In the second quadrant, it's $180^\circ-\theta$. In the third quadrant, it's $\theta-180^\circ$. In the fourth quadrant, it's $360^\circ-\theta$.
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