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Trigonometry |
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Trigonometric
functions of arcs from 0
to ±
2p
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Trigonometric
functions of negative arcs or angles |
Trigonometric
functions of complementary angles |
Trigonometric
functions of supplementary angles |
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Trigonometric
functions of arcs from 0
to ±
2p
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Trigonometric
functions of negative arcs or angles
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We say that arcs x
and x′
are opposite if |
x
+ x′
= 0 or x′
= -x.
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Comparing the corresponding sides of the congruent |
right-angled triangles in
the right figure, |
OPxP
and OPxP′ |
OSxS1
and OSxS1′ |
OSyS2
and OSyS2′ |
follows that we can express trigonometric functions of
a |
negative arc
(-x) by corresponding function of opposite arc |
x, that is |
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Example:
Given trigonometric functions of negative angle, arc or number should be expressed by the
same functions of the positive angle, arc or
number.
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a) sin
(-200°),
b) cos (-14p/3),
c) tan (-11),
d) cot (-750°).
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Solution: First express the angle
a
in decimal degrees, i.e. |
a) sin
(-200°)
= -
sin 200°
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c) tan
(-11)
= -
tan11 = -
tan (3p
+ 1.575222...) = -
tan 1.575222...
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d) cot
(-750°)
= -
cot 750° = -
cot (2 · 360° + 30°) = -
cot 30°.
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Trigonometric
functions of complementary angles
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Two angles, x
and p/2
-
x which form the right angle, are said to be complementary.
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Thus, comparing the
corresponding sides of the congruent right-angled triangles in
the below figure,
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OPxP
and OPx′P′,
OSxS1
and OSyS2′
and OSyS2
and OSxS1′
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Example:
The trigonometric functions of the given angle or arc should be expressed by corresponding function of the complementary angle.
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a) sin
30°,
b) cos (p/2
-
p/3),
c) tan 1,
d) cot 530°.
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Solution:
a) sin
30°
= cos(90° -
30°)
= cos 60° |
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d) cot 530° =
tan (90° -
530°) = tan (-
440°) = -
tan 440°
=
-
tan (2
· 180° +
80°)
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= -
tan 80° = -
cot (90°
-
80°)
= -
cot 10°.
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Example:
Simplify the expression
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Solution:
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Trigonometric
functions of supplementary angles
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Two angles,
x and
p
-
x, which
when added form a straight |
angle, are said to
be supplementary. |
Comparing the corresponding sides of the congruent right-angled
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triangles
in the right figure,
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Px′P′
= PxP => |
sin
(p
-
x) = sin x
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OPx′
= -OPx
=> |
cos
(p
-
x) = -
cos
x
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SxS1′
= -SxS1
=> |
tan
(p
-
x) = - tan
x
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SyS2′
= -SyS2
=> |
cot
(p
-
x) =
- cot x
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Example:
Trigonometric functions of a given arc, angle or number should be expressed by the corresponding function of the supplementary angle.
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a) sin
5p/6,
b) cos (-320°),
c) tan (p
-
1),
d) cot 30°.
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Solution:
a) sin
5p/6
= sin (p
-
5p/6)
= sin p/6 |
b) cos (-320°)
= cos (180° -
500°)
= -
cos 500°
= -
cos (360°
+ 140°)
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= cos (180° -
140°)
= cos 40°
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c) tan (p
- 1)
= -
tan 1
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d) cot 30°
= -
cot (180° -
30°)
= -
cot 150°.
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Trigonometry
contents A |
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