Wondering exactly what sine is.

[repugno]

I was wondering exactly what sine is. i know that the relationship between sine and the angle is expressed as sinx=opposite/hypotoneuse. But is sine somekind of constant you multiply the angle with to get the O/H ratio? How did the people calculate the ratios without calculaters when they only knew the angles?

 

[gazzo]

Can't sine theta be thought of as the y value of a point on the unit circle when you rotate it through theta?
cosine is the x value, tan is y/x.

Hence, sine 90 is 1, cos 90 is zero. tan 90 is indeterminant (1/0)

-Gareth

 

[Hurkyl]

How did the people calculate the ratios without calculaters when they only knew the angles?

Use algebra to find a lot of values, then interpolate to fill in the gaps.

You should be able to do sin 30 and sin 45 without a calculator. sin 72 can be done with pentagons. (same with cosine) Every multiple of 3 can then be gotten through the addition and subtraction of angles formulae. The half angle formuale can get you even finer... (and the one/third angle formulae, if you know the cubic formula)

One way your calculator could compute them is through their MacLauren series... but I understand they actually use a much cleverer method.

tan 90 is indeterminant (1/0)

No, tan 90 is undefined -- 90 is not in the domain of the tangent function.

 

[poolwin2001]

you can learn the values of some standard angles

degree 0    30       45                      60                   90
 Sin     0    1/2      1/squareroot 2     (root 3)/ 2         1
Cos    reverse the series

 

[gazzo]

No, tan 90 is undefined -- 90 is not in the domain of the tangent function.

Oops! Thanks Hurkyl