Special values of trigonometric functions

Trigonometry: Angles with sine, cosine and tangent

Special values of trigonometric functions

We have seen that the unit circle is symmetrical and that we therefore only need to know the first eighth. We will now look at special values for the first quarter. We then see the symmetry in the sine and cosine as we saw earlier. It is important to memorize these values.

#{\bf \alpha}# (in radians)	#{\bf 0}#	#{\bf \dfrac{\pi}{6}}#	#{\bf \dfrac{\pi}{4}}#	#{\bf \dfrac{\pi}{3}}#	#{\bf \dfrac{\pi}{2}}#
#{\bf \sin(\alpha)}#	#0#	#\dfrac{1}{2}#	#\dfrac{1}{\sqrt{2}}#	#\dfrac{\sqrt{3}}{2}#	#1#
#{\bf\cos(\alpha)}#	#1#	#\dfrac{\sqrt{3}}{2}#	#\dfrac{1}{\sqrt{2}}#	#\dfrac{1}{2}#	#0#
#{\bf \tan(\alpha)}#	#0#	#\dfrac{\sqrt{3}}{3}#	#1#	#\sqrt{3}#	-

Mnemonic

The values of the sine are successively: #\tfrac{1}{2}\sqrt{0}#, #\tfrac{1}{2}\sqrt{1}#, #\tfrac{1}{2}\sqrt{2}#, #\tfrac{1}{2}\sqrt{3}#, #\tfrac{1}{2}\sqrt{4}#.

For the cosine they are in reverse order.

We find the tangent with the formula:

\[\tan(\alpha)=\frac{\sin(\alpha)}{\cos(\alpha)}\]

Determine #\sin(\frac{5 \pi}{3})# without using your calculator.

#\sin(\frac{5 \pi}{3})=# #-\frac{1}{2}\sqrt{3}#

By use of reflections across the #x#-axis, we find #\sin(\frac{5 \pi}{3})=-\sin(\frac{\pi}{3})=-\dfrac{\sqrt{3}}{2}#.

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