Even and Odd Trigonometry Functions
All functions, including trigonometry
functions, can be described as being even, odd, or neither.
A function is said to be odd if
and only if f(-x) = - f(x) at all values of x and is symmetric with respect to
the origin.
Similarly, function is even if
and only if f(-x) = f(x) at all values of x and is symmetric to the y axis.
It is helpful to know if a function is odd or
even when we are trying to simplify an expression when the variable inside the
trigonometric functions are negative.
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Sin( -Ө ) = - sin Ө
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Cosec
( -Ө ) = - cosec Ө
|
|
Cos ( -Ө ) = cos Ө
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Sec (-Ө ) = sec Ө
|
|
Tan ( -Ө ) = - tan Ө
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Tan ( -Ө ) = - tan Ө
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Proof of Even Odd
Trigonometric Function:
From the definition of Cosine(Cos)
and Sine(Sin) in Unit Circle,
Put x = CosӨ
and y = SinӨ,
We can see that for both Ө and –Ө in the above
given figure. Value of “x” remains same, Hence Cos(Ө) = Cos(-Ө).
But, it is clearly seen that the value of y for Ө and – Ө are different an
additive inverse , thus Sin(Ө) is not equals Sin(-Ө), Here Sin(-Ө)= -Sin Ө
Similarly we can prove other trigonometric
functions,
|
Tan(-Ө)
|
=
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Sin(-Ө)
|
=
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-Sin(Ө)
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=
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-Tan(Ө)
|
|
Cos(-Ө)
|
Cos(Ө)
|
|||||
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Cot(-Ө)
|
=
|
1
|
=
|
1
|
=
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-Cot(Ө)
|
|
Tan(-Ө)
|
-TanӨ
|
|||||
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Cosec(-Ө)
|
=
|
1
|
=
|
1
|
=
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-Cosec(Ө)
|
|
Sin(-Ө)
|
-Sin(Ө)
|
|||||
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Sec(-Ө)
|
=
|
1
|
=
|
1
|
=
|
Sec(Ө)
|
|
Cos(-Ө)
|
Cos(Ө)
|
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