Acute Angles in ``Special'' Right Triangles

Recall that in a $30^\circ-60^\circ-90^\circ$ triangle, the length of the hypotenuse is twice that of the side opposite the 30 degree angle, and the side adjacent to the 30 degree angle is $\sqrt{3}$ times the side opposite the 30 degree angle.

In a $45^\circ-45^\circ-90^\circ$ triangle, the two legs are of equal length, and the hypotenuse is $\sqrt{2}$ times that length.

These relationships, when applied to the right triangle definitions of the trig functions, allow us to easily find and express the exact value of any trig function of such an angle.

The following table summarizes these special values for the special angles in QI (i.e. between 0 and $\pi/2$ radians) so the same applies to integer multiples of $2\pi$ of these angles. For special angles in QII through QIV, simply determine the desired trig value of the reference angle, then assign the correct sign (plus or minus) according to the quadrant in which the terminal side lies.

A consequence of these special angle relationships is, given one trigonometric functional value of such an angle, the other five are easily determined from the right triangle definitions and the Pythagorean Theorem.

$\theta$	$\sin\theta$	$\csc\theta$	$\cos\theta$	$\sec\theta$	$\tan\theta$	$\cot\theta$
$\frac{\pi}{6},\;$or$\;30^\circ$	1/2	2	$\sqrt{3} / 2$	$(2\sqrt{3}) / 3$	$\sqrt{3} / 3$	$\sqrt{3}$
$\frac{\pi}{4},\;$or$\;45^\circ$	$(\sqrt{2}) / 2$	$\sqrt{2}$	$(\sqrt{2}) / 2$	$\sqrt{2}$	1	1
$\frac{\pi}{3},\;$or$\;60^\circ$	$\sqrt{3} / 2$	$(2\sqrt{3}) / 3$	1/2	2	$\sqrt{3}$	$\sqrt{3} / 3$