“Oh,” says the student, “I’ll just put it in my calculator.”

I fold my arms.

”$\sin\left(\frac{\pi}{6}\right) = 0.00914$!” he says, confidently.

I sigh.

“Oh!” he continues, brightly, “is it meant to be in radians? I don’t like radians.”

This is the problem with calculators

Now, I love calculators. Machines that do complicated sums for you, much more accurately than you can alone. Terrific invention. Wouldn’t be without one.

However, using them without thinking is worse than not using them. (I once had a student who insisted Mars was 5cm away because he’d plugged all the values into the formula and that’s what his calculator said and no amount of ‘but 5cm is about that far!’ would persuade him otherwise).

So, if you want to have a shot at doing well with trig, you need to understand what’s going on with sine and cosine (also tangent, to a lesser degree*).

The sine function.

Here’s what sine does: it takes an angle and tells you how many metres above the ground you would be if you flew off for a metre in that direction, ignoring gravity. For instance, if you took off at $45^\circ$** (which is, of course, $\frac{\pi}{4}$ radians), after a metre’s travel, you’d be $1 \times \sin(45^\circ)$ metres off of the ground. (It works out to be $\frac{\sqrt 2}{2}$, which is about 0.707 – as well you know.)

If you took off at $120^\circ$ ($\frac{2\pi}{3}$ radians) above the ground, you’d be flying backwards, because 90 degrees is straight up, but you’d still go up – this time, if you travel one metre, you’ll go up $1 \times \sin(120^\circ)$, which is $\frac{\sqrt{3}}{2}$, or 0.866 in terrible decimals.

Now, how about something like $210^\circ$ ($\frac{7\pi}{6}$ radians)? That’s backwards and slightly downwards, so you end up below where you started – with a negative height. The calculator spews out $\sin(210^\circ) = -\frac{1}{2}$.

Finally, let’s look at $315^\circ$ ($\frac{7\pi}{4}$), which isn’t far off of a full circle. You’re travelling forwards again, but you’ve not quite done a full backflip, so you’re still heading downwards. $\sin(315^\circ) = -\frac{\sqrt{2}}{2}$.

Hm, funny, that: 315º is 45º short of a full circle, and $\sin(315^\circ)$ is just the negative of $\sin(45^\circ)$. Coincidence? I think not. But I’ll deal with that in part II.

The cosine function.

Cosine is similar but slightly different: it tells you how far forwards you go – as in, how far you travel horizontally rather than vertically. For instance, if you took off at $45^\circ$** (which is, of course, $\frac{\pi}{4}$ radians), after a metre’s travel, you’d be $1 \times \cos(45^\circ)$ metres forward from where you started. (That works out to be $\frac{\sqrt 2}{2}$, which is about 0.707.)

If you took off at $120^\circ$ ($\frac{2\pi}{3}$ radians) above the ground, you’d be flying backwards, because 90 degrees is straight up, but you’d still go up – this time, if you travel one metre, you’ll go up $1 \times \cos(120^\circ)$, which is $-\frac{1}{2}$: you’ve gone half a metre backwards.

Now, how about something like $210^\circ$ ($\frac{7\pi}{6}$ radians)? That’s backwards and slightly downwards, so you still end up behind where you started. The calculator spews out $\cos(210^\circ) = -\frac{\sqrt{3}}{2}$.

Finally, let’s look at $315^\circ$ ($\frac{7\pi}{4}$), which isn’t far off of a full circle. You’re travelling forwards again, but you’ve not quite done a full backflip, so you’re still heading downwards. $\cos(315^\circ) = \frac{\sqrt{3}}{2}$.

Hm, funny, that: 315º is 45º short of a full circle, and $\cos(315^\circ)$ is the same as $\cos(45^\circ)$. Coincidence? I think not. But I’ll deal with that in part II.

Summary

To figure out the correct sign of a sine or a cosine, it’s easy enough: you think about which directions you’d fly if you took off at that angle. Forwards or backwards? Upwards or downwards? After that, it’s easy.

More on sine and cosine another time…

* no pun intended. ** Degrees are rubbish, but for now I’ll use them to avoid confusion.