In this discussion, we are going to prove the Odd and Even trig Identities.
Here is an outline of this discussion:
The Definition of an Odd and Even Function
Proving Sine is an Odd Function
Proving Cosine is an Even Function
Proving Tangent is an Odd Function
Proving Cotangent is an Odd Function
Proving Cosecant is an Odd Function
Proving Secant is an Even Function
What does it mean for a function to be odd or even?
The Definition of an Odd and Even Function
Recall the definition of an odd and even function:
A function, f is odd if f(-theta) = -f(theta).
A function, f is even if f(-theta) = f(theta).
We will use the definition to prove the odd/even trig identities, but first, let us begin with a circle.
Given a circle with radius r, let theta be an arbitrary angle drawn in the positive direction from the positive x-axis. You can draw the angle anywhere you like, but for convenience, I draw the angle in quadrant one WLOG.
Since I choose to draw theta in quadrant one, I simply draw -theta in quadrant four. It is important to note that theta and -theta have same magnitude. The only difference is that -theta is measured in the opposite direction of theta from the positive x-axis.
Let theta have terminal points (x,y).
Then -theta has terminal points (x,-y).
Proving sine is an odd Function
Let us prove that sine is an odd function using the circle and the definition of an odd function.
We know that
sin(theta) = y/r
Then
sin(-theta)
=
-y/r
=
-(y/r)
=
-(sin(theta))=
-sin(theta)
Thus,
sin(-theta) = -sin(theta),
which means that sine is an odd function by the definition.
Proving Cosine is an Even Function
Next, we prove that cosine is an even function.
We know that cos(theta) = x/r
Then cos(-theta) = x/r = cos(theta).
Thus, cos(-theta) = cos(theta), which means that cosine in an even function by the definition of an even function.
Next, let us prove that tangent is an odd function.
Proving Tangent is an Odd Function
We know that tan(theta) = y/x.
Then
tan(-theta)
=
sin(-theta)/cos(-theta)
=
-sin(theta)/cos(theta)
=
-tan(theta)
Thus,
tan(-theta) = -tan(theta), which means that tangent is an odd function.
Proving Cotangent is an Odd Function
Now let us prove that cotangent is an odd function.
We know that
cot(theta) = x/y.
Then
cot(-theta)
=
cos(-theta)/sin(-theta)
=
cos(theta)/-sin(theta)
=
-cot(theta)
Thus,
cot(-theta) = -cot(theta), which means that Cotangent is an odd function.
Proving Cosecant is an Odd Function
Let us prove that cosecant is an odd function.
We now that
csc(theta) = 1/sin(theta)
Then
csc(-theta)
=
1/sin(-theta)
=
1/-sin(theta)
=
-csc(theta)
Thus,
csc(-theta) = -csc(theta), which means that cosecant is an off function.
Proving Secant is an Even Function
Last, we prove that secant is an even function.
We know that
sec(theta) = 1/cos(theta).
Then,
sec(-theta)
=
1/cos(-theta)
=
1/cos(theta)
=
sec(theta)
Thus,
sec(-theta) = sec(theta), which means that secant is an even function.
Thank you for reading!
I hope that I can assist you on your math journey!
Sincerely,
➕MathAngel369➕
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