how do you do arcsin 1/2 without a calculator

Thanks anon

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I always start these with an equilateral triangle of side length 2.    Angles will be 60 degrees

I then put in a height which will split the base in half.

Now you have 2 right angled triangles with hypotenuse=2 and base is 1 unit

using Pythagoras you can easily determine that the height is   $${\sqrt{{\mathtt{3}}}}$$ units.

The angles in each one will be 90,60 and 30 degrees.

now you can see that   sin(30) = 1/2

 

I draw this either on paper or in my head EVERY time.

arcsin (1/2) means

$$Sin\theta=\frac{1}{2}$$

Recall that the sin function involves the ratio of the opposite side divided by the hypotenuse. So the opposite side is half of the hypotenuse. If you draw a 30-60-90 triangle with the hypotenuse equal to 1, you can determine that the side opposite the 30 degree angle is half as long.