Even and Odd Trigonometry Functions

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.

Sin( -Ө ) = - sin Ө	Cosec ( -Ө ) = - cosec Ө
Cos ( -Ө ) = cos Ө	Sec (-Ө ) = sec Ө
Tan ( -Ө ) = - tan Ө	Tan ( -Ө ) = - tan Ө

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)	=	Sin(-Ө)	=	-Sin(Ө)	=	-Tan(Ө)
		Cos(-Ө)		Cos(Ө)
Cot(-Ө)	=	1	=	1	=	-Cot(Ө)
		Tan(-Ө)		-TanӨ
Cosec(-Ө)	=	1	=	1	=	-Cosec(Ө)
		Sin(-Ө)		-Sin(Ө)
Sec(-Ө)	=	1	=	1	=	Sec(Ө)
		Cos(-Ө)		Cos(Ө)

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Cofunction/Confunction Identities of Trigonometry

Cofunction Identities of Trigonometry

The cofunction identities of trigonometry define the relationship between sine (Sin), cosine (Cos), tangent (Tan), cotangent (Cot), secant (Sec) and cosecant (Cosec). The value of a trigonometric function at an angle equals the value of the cofunction of the complement.(A complement is defined as two angles whose sum is 90°)

In Radians:

Sine and cosine are cofunctions and complements	Sin	(	π	-	Ө	)	=	CosӨ	Cos	(	π	-	Ө	)	=	SinӨ
			2								2
Tangent and cotangent are cofunctions and complements	Tan	(	π	-	Ө	)	=	CotӨ	Cot	(	π	-	Ө	)	=	TanӨ
			2								2
Secant and cosecant are cofunctions and complements	Sec	(	π	-	Ө	)	=	CosecӨ	Cosec	(	π	-	Ө	)	=	SecӨ
			2								2

In Degree:

Sine (Sin) and cosine (Cos) are cofunctions and complements	sin(90° - Ө) = cos Ө	cos(90° - Ө) = sin Ө
Tangent (Tan) and cotangent(Cot) are cofunctions and complements	tan(90° - Ө) = cot Ө	cot(90° - Ө) = tan Ө
Secant(Sec) and cosecant(Cosec) are cofunctions and complements	sec(90° - Ө) = csc Ө	csc(90° - Ө) = sec Ө

If you find any doubts please free to ask them in the comment section, We would be happy to reply with solution ASAP.