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Trigonometry |
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Trigonometric
functions of arcs from 0
to ±
2p
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Trigonometric
functions of arcs that differ on p/2 |
Trigonometric
functions of arcs that differ on p |
Trigonometric
functions of arcs whose
sum is
2p |
Trigonometric
functions values and identities examples |
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Trigonometric
functions of arcs that differ on p/2
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Comparing the corresponding sides of the congruent right-angled |
triangles,
in the right figure, we get the relations of trigonometric
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functions of
an arc x and
the arc p/2
+ x |
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Px′P′
= OPx => |
sin
(p/2
+ x) = cos x
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OPx′
= -PxP
=> |
cos
(p/2
+ x) = -sin
x
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SxS1′
= -SyS2
=> |
tan
(p/2
+ x) = -cot x
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SyS2′
= -SxS1 => |
cot
(p/2
+ x) = -tan x
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Example:
Trigonometric functions of a given arc, angle or number should be expressed by
the corresponding function of angle which differ from the given for
90° (p/2).
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a) sin
1,
b) cos 150°,
c) tan (-7p/4),
d) cot 50°.
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Solution:
a) sin
1 = -
cos (p/2
+ 1)
= -
cos 2.570796... |
b) cos 150°
= cos
(90°
+ 60°) = -
sin 60°
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c) tan (-7p/4)
= -
cot (p/2
-
7p/4)
= -
cot ( -
5p/4)
= cot
(p
+ p/4) =
cot
p/4
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d) cot 50°
= -
tan
(90°
+ 50°) = -
tan 140°.
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Trigonometric
functions of arcs that differ on p
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Comparing the corresponding sides of the congruent right-angled |
triangles,
in the right figure, we get the relations of trigonometric
|
functions
of an arc x and
the arc p
+ x |
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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 angle which differ from the given for
180° (p).
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a) sin
235°,
b) cos p/6,
c) tan (-300°),
d) cot 4.
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Solution:
a) sin
235°
= sin (180°
+ 55°) = -
sin 55°
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b) cos p/6
= -
cos (p
+ p/6) =
-
cos
7p/6
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c) tan (-300°)
= tan (180° -
300°) = tan (-120°)
= -
tan 120°
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d) cot 4
= cot (p
+ 0.858407...) = cot 0.858407....
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Example:
Simplify expression
cot (p
-
x)
· cos
(p/2
+ x)
+ tan (p/2
-
x)
· tan
(p
+ x) -
cos (-
x)
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Solution:
cot (p
-
x)
· cos
(p/2
+ x)
+ tan (p/2
-
x)
· tan
(p
+ x) -
cos (-
x) = |
= -
cot x
· ( -
sin x)
+ cot x
· tan x
-
cos x = cos x
+ 1 -
cos x =
1.
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Trigonometric
functions of arcs whose
sum is 2p
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The
right figure shows relations between sides of the congruent |
right-angled triangles as
follows,
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PxP′
= -PxP
=> |
sin
(2p
-
x) = -sin
x
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OPx
= cos (2p
-
x)
= cos x |
cos
(2p
-
x) = cos
x
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SxS1′
= -SxS1
=> |
tan
(2p
-
x) = -tan
x
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SyS2′
= -SyS2
=> |
cot
(2p
-
x) = -cot
x
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Trigonometric
functions values and trigonometric identities examples
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Example:
Trigonometric functions of a given arc, angle or number should be expressed by the corresponding function of angle which when added with a given make
360° (2p).
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a) sin
p/3,
b) cos 1,
c) tan 330°,
d) cot 10p/11.
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Solution:
a) sin
p/3
= -sin (2p
-
p/3) =
-
sin 5p/3 |
b) cos 1 = cos (2p
-
1) = cos 5.283185...
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c) tan 330° =
tan (360° -
330°) = -
tan 30°
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d) cot 10p/11
= -
cot (2p
-
10p/11) =
-
cot p/11.
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Example:
Prove that sin
320°
+ cos 50° = 0.
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Solution:
Since
sin 320° = sin (360°
-
40°) = -
sin 40°, and as
cos 50° = sin 40° |
then
-
sin 40°
+ sin
40° = 0.
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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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