How do you find the arcsin(sin((7pi)/6))?

1 Answer
Feb 10, 2015

The answer is:
arcsin(sin(7pi/6))=-pi/6.

The range of a function arcsin(x) is, by definition ,
-pi/2<=arcsin(x)<=pi/2
It means that we have to find an angle alpha that lies between -pi/2 and pi/2 and whose sin(alpha) equals to a sin(7pi/6).

From trigonometry we know that
sin(phi+pi)=-sin(phi)
for any angle phi.
This is easy to see if use the definition of a sine as an ordinate of the end of a radius in the unit circle that forms an angle phi with the X-axis (counterclockwise from the X-axis to a radius).
We also know that sine is an odd function, that is
sin(-phi)=-sin(phi).

We will use both properties as follows:
sin(7pi/6)=sin(pi/6+pi)=-sin(pi/6)=sin(-pi/6)

As we see, the angle alpha=-pi/6 fits our conditions. It is in the range from -pi/2 to pi/2 and its sine equals to sin(7pi/6). Therefore, it's a correct answer to a problem.