How do you evaluate #arcsin(1)# without a calculator?

Question

Josiah Anderson · Accepted Answer

#pi/2=90^o#

Arcsin is the reverse of the sine function - that is, it looks at a ratio and asks what angle can create that ratio.

Here we have #arcsin(1)#, which is asking what angle creates a sin ratio of 1.

We can express this sin ratio as #sinx="opp"/"hyp"=1#. So what angle can create an opposite side that has an equal length hypotenuse?

Look at the point where the line segment and circle meet. The line segment is the hypotenuse of a circle and the opposite is a line dropped from that point down on to the x-axis. If we move the point along the circle towards the y-axis, the opposite side gets longer and starts to approach the same length as the hypotenuse. It's at the point where the line segment stretches straight up the y-axis, and consequently the opposite side does as well, that we get the equal length needed. That angle is #pi/2=90^o#

Anna Andrews · Answer

undefined To evaluate $ \arcsin(1) $ without a calculator, recognize that the arcsin function returns an angle whose sine is the input. Since the sine function's range is $[-1, 1]$, $ \arcsin(1) $ will return an angle whose sine is 1. The only angle in the unit circle whose sine is 1 is $ \frac{\pi}{2} $ radians or $ 90^\circ $. Therefore, $ \arcsin(1) = \frac{\pi}{2} $ radians or $ 90^\circ $.