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| Trigonometry |
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Trigonometric
functions of arcs from 0
to ±
2p
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Trigonometric
functions of angles lying on axes |
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Trigonometric
functions values and identities examples |
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Trigonometric
functions of angles lying on axes
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Values of trigonometric functions of characteristic arcs,
0,
p/2,
3p/2
and
2p
follow directly from the definitions.
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Thus, for functions sine and
cosine from
the above figure we read the coordinates of the arc terminal point
P
that is, for the sine function we read the ordinate
while for the cosine function the abscissa of the terminal
point.
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Therefore,
sin 0 = 0,
sin p/2
= 1,
sin p =
0,
sin 3p/2
= -1,
sin 2p
= 0,
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and cos
0 = 1,
cos p/2 =
0,
cos p =
-1,
cos 3p/2 =
0,
cos 2p =
1.
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Point S1
whose ordinate determines the value of the
function tangent,
for the arcs,
0,
p
and 2p,
coincide with the initial point P1
of the arc, i.e., lies on the x-axis,
see the down figure.
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Thus, tan 0 = 0, tan p
= 0, tan 2p =
0, while for arcs,
p/2
and 3p/2
their terminal side or its extension lies on the y-axis, that is parallel with tangent
x =
1. There is no intersection S1
and we say that for these arcs the function tangent is
undefined.
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However, if we follow the intersection point
S1 while the arc increases from
0
to p/2
we see that it moves away the x-axis and its
ordinate tan a1
tends to infinity (+
oo) which can be written as,
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when a1
® p/2,
tan a1
® +
oo
or tan
p/2
= oo
.
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If we continue to follow changes of the values of the function tangent, i.e., changes of the ordinates of the
intersection
S1′ while the arc increases from
p/2
to p that is, while the terminal side of the angle
a2
or its extension continue rotates in the positive direction, we see that point
S1′ moves toward the
x-axis and at the same time its ordinate
tan
a2 increases from
-
oo to
0.
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Thus we can write, tan
p/2
= ± oo
. Examining the same way it follows that, tan
3p/2
= ± oo
.
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The intersection point S2,
whose abscissas determine the values of the function
cotangent, coincide with the point Pp/2 for the arcs
p/2
and 3p/2 on the
y-axis, so
cot p/2
= 0 and
cot 3p/2
= 0 while for arcs, 0
(2p) and
p, the terminal side of the
corresponding central angle, or its extension, lies on the
x-axis
parallel with the tangent
y = 1, so there is no intersection
point.
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We say that the function cotangent is undefined for those arcs.
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To determine bounds of the values that the function cotangent
takes while the terminal point of an arc rounds the unit circle in the
positive direction passing through mentioned characteristic values,
0(2p) and
p, we should follow the intersection point
S2
on the tangent
y = 1, i.e., the changes of its abscissas
cot
a,
see
the above figure.
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Trigonometric
functions values and identities examples
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| Example:
Calculate, sin
3p/2
· cos(-
p)
+ tan 5p/4.
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| Solution: sin
3p/2
· cos(-
p)
+ tan 5p/4
= -
1 · (-
1) + tan (p
+ p/4)
= 1 + tan p/4
= 1 + 1 = 2. |
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| Example:
Calculate, |
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Solution:
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Example:
Prove the identity,
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cos2 p/3
· sin (p/2
-
x) -
cos (p
-
x)
· cos2 p/6
= tan (p/2 +
x)
· sin (2p
-
x).
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Solution:
Since sin
(p/2
-
x) = cos
x,
cos (p
-
x) = -
cos
x, tan (p/2 +
x) = -
cot
x
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and sin
(2p
-
x) = -
sin x
then,
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(cos p/3)2
· cos
x -
(-
cos
x)
· (cos p/6)2
= -
cot
x
· (-
sin x),
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(1/2)2
· cos
x + (Ö3/2)2
· cos
x = (cos
x/sin x)
· sin x
=> cos
x = cos
x.
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Example:
Prove the identity,
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cot2 (p +
x)
· cos2 (p/2 +
x) + sin (-
x) · sin (p +
x) = tan (2p
-
x) · cot (-
x).
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Solution:
[cot (p +
x)]2
· [cos (p/2 +
x)]2 +
(-
sin x)
· sin (p +
x) = (-
tan x)
· (-
cot x),
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(cot
x)2
· (-
sin x)2
+ (-
sin x) · (-
sin x) = (sin x/cos
x)
· (cos
x/sin x)
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cos2
x + sin2
x = (sin x/cos
x)
· (cos
x/sin x) = 1.
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| Trigonometry
contents A |
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© 2004 - 2020, Nabla Ltd. All rights reserved. |
