Special Angles

Calculators stayed in bags at the boards today, and your group still produced exact trigonometric ratios — because two triangles you can draw from scratch contain all of them. Exact values like $\frac{\sqrt{3}}{2}$ are not decimals that got dressed up; they are what the triangles actually say, before rounding throws anything away.

Two triangles, one table

Cut a unit square along its diagonal: a right triangle with two 45° angles, legs 1 and 1, hypotenuse $\sqrt{2}$. Cut an equilateral triangle of side 2 down its altitude: a right triangle with angles 30° and 60° and sides 1, $\sqrt{3}$, 2. Every entry below is just opposite, adjacent, and hypotenuse read from one of those two pictures:

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$
$0°$	$0$	$1$	$0$
$30°$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$45°$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$60°$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$90°$	$1$	$0$	undefined

Do not memorise this table — rebuild it. Sketch the two triangles in your margin at the start of any trig work; thirty seconds of drawing beats a term of flashcards, and it is exactly the kind of note your future forgetful self will thank you for.

Past ninety degrees

Put a rotating arm of length 1 at the origin and let $\theta$ be its angle from the positive $x$-axis. The arm’s tip has coordinates $(\cos \theta ,\sin \theta )$ — and now every angle from 0° to 360° has ratios, right angle or not. An angle past 90° borrows its numbers from a related special angle in the first quadrant; only the signs change, following the coordinates. So $\sin 150°=\sin 30°=\frac{1}{2}$ (height still positive in the second quadrant), while $\cos 135°=-\frac{\sqrt{2}}{2}$ (the arm now points left).

One consequence worth sitting with: a single ratio value belongs to two angles in a full turn. If $\sin \theta =\frac{1}{2}$, then $\theta =30°$ or $\theta =150°$ — the arm reaches the same height on the way up and on the way around. That doubling has consequences for The Sine Law, and it makes lovely Estimation Duels material: where must $\sin 50°$ sit relative to $\sin 45°$ and $\sin 60°$?

Exact-value fluency is the opening act of Trig Ratios and Laws Practice.

Curriculum connection

D1.1

determine the exact values of the sine, cosine, and tangent of the special angles: $0^{\circ }$, $30^{\circ }$, $45^{\circ }$, $60^{\circ }$, and $90^{\circ }$

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D1.2

determine the values of the sine, cosine, and tangent of angles from $0^{\circ }$ to $360^{\circ }$, through investigation using a variety of tools (e.g., dynamic geometry software, graphing tools) and strategies (e.g., applying the unit circle; examining angles related to special angles)

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D1.3

determine the measures of two angles from $0^{\circ }$ to $360^{\circ }$ for which the value of a given trigonometric ratio is the same

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