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

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

We can express this sin ratio as

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

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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 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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