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| Trigonometry |
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Basic relationships
between trigonometric functions of the same angle |
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Basic relationships
between trigonometric functions of the same angle shown in the
unit circle |
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Basic relationships
between trigonometric functions of the same angle, examples |
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Basic relationships
between trigonometric functions of the same angle
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Using the Pythagorean theorem for the right triangle
in the
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figure we get the fundamental trigonometric identity
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which, because of periodicity of the trigonometric functions,
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holds for an arbitrary angle
x =
a +
k · 2p
therefore,
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sin2
x
+ cos2 x = 1,
x
Î
R.
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From this identity and the definitions of the functions, tangent
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we can derive twelve formulas so that each function is expressed
through another three. Thus,
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By plugging (1) and (2) into,
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Dividing the basic identity by cos2 x,
sin2
x
+ cos2 x = 1 | ¸ cos2 x
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Dividing the basic identity by sin2
x,
sin2
x
+ cos2 x = 1 | ¸
sin2
x
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Basic relationships
between trigonometric functions of the same angle shown in the
unit circle
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Given relations between the trigonometric functions of the same angle, expressed by absolute value,
are included in the definitions of the functions in right triangles shown in
Fig. a - d, and included in the table below.
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Basic relationships
between trigonometric functions of the same angle, examples
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| Example:
Find values of other trigonometric functions of
an angle a,
if given sin a
= -
4/5 and
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270° < a
< 360°.
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Solution:
Since a
is a forth-quadrant angle, then
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| Example:
Find the value
of |
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Solution:
Dividing nominator and denominator by
cos2 x,
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| Example:
Prove the
identity |
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| Solution: |
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| Example:
Given
sin x + cos x
= a, find
sin4
x
+ cos4 .
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Solution:
Since sin2
x
+ cos2 x = 1 then,
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(sin2
x
+ cos2 x)2 = sin4
x
+ cos4 + 2sin2
x · cos2 x
= 1 or
sin4
x
+ cos4 = 1
-
2sin2
xcos2 x.
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As given sin
x + cos x = a then,
(sin x + cos x)2 =
a2
or sin2
x
+ cos2 x
+ 2sin
x cos x = a2
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therefore, sin
x
· cos x =
(a2
-
1)/2 and
it follows that sin4
x
+ cos4 = 1
- (a2
-
1)2/2.
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| Pre-calculus contents
F |
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