Why does the progression of values of $\sin(x)$ seem so arbitrary?

Question

I don't know how to ask this in a way that doesn't sound a little stupid, but what I mean is, the progression of values of say, sin(x) seems really arbitrary. Is there an intuitive way to understand why, for example:

   x            y
-------   -------------
sin(45) = 0.85090352453
sin(46) = 0.90178834764

What is determining the amounts that Y goes up or down by as X changes? And even better, is there something (fairly accessible to non-mathematicians) I can read that can explain this sort of thing in a very intuitive way? I don't want to prove that it works, just understand how come. Thanks!

Answer 1

Do you mean to be working in radians, because $\sin 45$ is a little bit arbitrary. So, if $\sin 45$ seems arbitrary, it is because $45$ is a bit arbitrary in the units you have chosen.

Lets explain what all of this means.

45 represents the point you would be if you traveled along the circumference of the circle in a counter clockwise direction a distance 45 units.

That is 7 full revolutions plus about 1/6 an additional revolution.

$45 = 14\pi + 1.02$

I have marked approximately where 45 and 46 lie on the unit circle.

$\sin 45$ is the distance that point is above the $x$ axis. $\cos 44$ is the distance to the right of the $y$ axis.

$\sin 46$ is nearly the same height above the $x$, axis.

Other things that make $\sin x$ appear a little arbitrary... Except of $0$ if $x$ is rational $\sin x$ will irrational (transcendental in fact).

Some other rules you can live by.

$-1\le \sin x \le 1$

$\sin (x+\pi) = -\sin x\\ \sin -x = -\sin x\\ \sin (x+2\pi) = \sin x$

$|\sin x - \sin y| \le |x-y|$

You can find $\sin x$ by plugging into this formula.

$\sin x = \sum_\limits{n=0}^\infty (-1)^n\frac {x^{2n+1}}{2n+1!}$ Which works very well for small values of $x$ but would take a long time to converge for a vale of $x$ as big as $45.$

but since $\sin 45 =\sin 1.02$ you can plug $1.02$ into the formula and it will converge much faster.

I hope this helps.

If we are working in degrees:

$46^\circ$ is much closer to $45^\circ$ than we were when we were working in radius.

$360^\circ = 2\pi$ radians

or 1 degree $= \frac {\pi}{180}$ radians.

or 1 radian $\approx 57$ degrees.

$45^\circ$ is a special angle. We can use it to construct an isosceles right triangle and find $\sin 45^\circ$ exactly

$\sin 45^\circ = \frac {\sqrt {2}}{2} \approx 0.707$

Most other angles are still problematic.

$\sin 46^\circ \approx 0.719$

since the angle is small. If we travel from $(\cos 45^\circ, \sin 45^\circ)$ tangent to the circle the distance equivalent to 1 degree, we will arrive very nearly at $(\cos 46^\circ, \sin 46^\circ)$

The slope of the tangent is the negative reciprocal of the slope of the radius.

We travel vertically $\cos 45(\frac {\pi}{180})$ and to the left $\sin 45(\frac {\pi}{180})$

In degrees $|\sin (x+1) - \sin x| \le \frac {\pi}{180}$

Answer 2

It seems that you are using radians rather than degrees. Degrees are most common in day to day use (a full circle is $360$). Serious maths usually uses radians (a full circle is $2 \pi$). Calculators that support sine usually offer both but the default varies. If you did not intend to use radians then check the instructions for a degree mode. Computing languages also vary but radians are common. An option for degrees is less common but it can be handled. If $\theta$ is your angle in degrees then use $\sin(\theta \pi / 180)$.

As Doug M explains , $1$radian is a quite large and fairly arbitrary amount, around $57.3$ degrees. He explains the consequences well.

If you switch to degrees then it will look quite different. $45$ is quite special in degrees and its $\sin$ is $\frac{\sqrt(2)}{2}$ which is approximately $0.7071$. $1$ degree is fairly small. If you calculate $\sin$ for $45, 46, 47, ...$ then the pattern should be clearer. It will grow approximately at a constant rate but gradually slowing until it reaches $90$ when it will reach its maximum of $1$ and then it will start to fall.

Alternatively, go back to radians but use a smaller step e.g. $0.01, 0.02, 0.03, ...$. Again, you will see a pattern.