Hoofdstuk 11 Goniometrie *: Graden en Radialen *
Special values of trigonometric functions
The values of cosine and sine for some special cases are provided in the table below.
|
#\alpha# (in degrees) |
#0^\circ# |
#30^\circ# |
#45^\circ# |
#60^\circ# |
#90^\circ# |
|
#\alpha# (in radians) |
#0# |
#\dfrac{\pi}{6}# |
#\dfrac{\pi}{4}# |
#\dfrac{\pi}{3}# |
#\dfrac{\pi}{2}# |
|
#\cos(\alpha)# |
#1# |
#\dfrac{\sqrt{3}}{2}# |
#\dfrac{1}{\sqrt{2}}# |
#\dfrac{1}{2}# |
#0# |
|
#\sin(\alpha)# |
#0# |
#\dfrac{1}{2}# |
#\dfrac{1}{\sqrt{2}}# |
#\dfrac{\sqrt{3}}{2}# |
#1# |
Because of the theory Right triangles and trigonometric functions the formulas can be derived from examining a triangle #ABC# with right angle #B# and angle #\alpha# in #A#.
In the case #\alpha = 0# is #|BC| = 0#, so the triangle degenerates into a line segment. Hence #\sin(0) = \dfrac{|BC|}{|AC|} = 0#. Moreover #|AC|=|AB|# applies, such that #\cos(0) = \dfrac{|AB|}{|AC|} = 1#.
In the case #\alpha=30^\circ# we are dealing with a triangle of which side #|BC|# is half the length of the long side. This determines #\sin(30^\circ)#. The cosine follows from the Pythagorean theorem. The third angle (in #C#) is equal to #60^\circ#, hence in that case the cosine and sine interchange values in relation to #30^\circ#.
In the case #\alpha = 45^\circ# we have #|AB|=|BC|#, of which it follows that cosine and sine are equal. The values of cosine and sine then are a direct result of the Pythagorean theorem.
All angles in the table are acute, but as we saw earlier, the values of other angles can easily be derived from these.
Since #135^\circ = 180^\circ - 45^\circ#. It follows from the theory Periodicity of trigonometric functions that #\displaystyle\sin(135^\circ)= \sin (45^\circ) = \frac{1}{2}\sqrt{2}#.
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