11.4.1 Measuring Triangles

This subsection is not completed yet, but in the meantime here are some practice questions.

The special angles that you need to know, with their sines, cosines, and tangents are:

$\theta$ $\sin\theta$ $\cos\theta$ $\tan\theta$
$30^\circ=\pi/6$ $\dfrac{\sqrt1}{2}$ $\dfrac{\sqrt3}{2}$ $\dfrac{\sqrt1}{\sqrt3}=\dfrac{1}{\sqrt3}$
$45^\circ=\pi/4$ $\dfrac{\sqrt2}{2}$ $\dfrac{\sqrt2}{2}$ $\dfrac{\sqrt2}{\sqrt2} =1$
$60^\circ=\pi/3$ $\dfrac{\sqrt3}{2}$ $\dfrac{\sqrt1}{2}$ $\dfrac{\sqrt3}{\sqrt1}=\sqrt3$

When written in this way, the sines make a lovely pattern of $\sqrt1/2$, $\sqrt2/2$, $\sqrt3/2$, with the cosines in the opposite order. You can work out the tangents by dividing sine by cosine. (I’m using $\sqrt1/2$ to make the pattern clear, but we would of course use $1/2$ instead.)

Alternatively, you can work it out by sketching the special triangles: Diagram: special-angles 1

Or if you want to use a calculator, you can learn to recognise $0.866\cdots$ as $\sqrt3/2$ and $0.707\cdots$ as $\sqrt2/2$, or you can get the exact value by squaring the answer, for example $\sin60^\circ$ will give $0.866\cdots$, and if you square this, you’ll get $3/4$, whose square root is $\sqrt3/2$.

Show practice questions

Add these questions to my cards
About practice questions