# Quod Erat Demonstrandum

## 2010/05/13

### Trivial sharing on trigonometry

Filed under: Additional / Applied Mathematics,mathematics,NSS — johnmayhk @ 7:33 上午
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What’s wrong with the following steps?

$\sin\theta < 0$

$\Rightarrow \theta$ lies in the 3rd or 4th quadrant

$\Rightarrow 180^o < \theta < 360^o$

$\Rightarrow 90^o < \frac{\theta}{2} < 180^o$

$\Rightarrow \sin\frac{\theta}{2} > 0$

For questions like: “If $\sin\theta = -\frac{1}{2}$, evaluate $\sin\frac{\theta}{2}$“, we need to consider the sign of $\sin\frac{\theta}{2}$.

As you may see that the incorrect steps above occur at

$\theta$ lies in the 3rd or 4th quadrant

$\Rightarrow 180^o < \theta < 360^o$

A correct expression is suggested as

$\theta$ lies in the 3rd or 4th quadrant

$\Rightarrow \theta = 360^on + \alpha$ (for some integer n and $180^o < \alpha < 360^o$)

Hence $\sin\frac{\theta}{2} > 0$ may NOT be always true.

Urm… I’d made similar mistake too, the sharing is a reminder to me… ^_^"

## 4 則迴響 »

1. That’s the mistake i made in the test, i think i hadn’t thought about the possibility of sinX/2 would be negative in the test.

迴響 由 Matthew — 2010/05/13 @ 8:05 下午 | 回應

• Not only you, some students and even I had made the mistake too! And the mistake made you lost 1 mark, otherwise you’ll obtain full mark in the quiz! Good job Matthew!!

迴響 由 johnmayhk — 2010/05/13 @ 10:49 下午 | 回應

2. thanks

迴響 由 Matthew — 2010/05/13 @ 11:02 下午 | 回應

3. Just a little bit following up.

Perhaps we should be careful in proving that $\sin x$ and $\tan \frac{x}{2}$ are of the same sign for any real number x.

迴響 由 johnmayhk — 2010/05/18 @ 2:21 下午 | 回應