Question

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

Answer 1

`pi/2=90^o`

Explanation:

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?

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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`