Which trigonometric function is an odd function and why?

The cosine function is odd because it is represented by the x-coordinate of the points on the unit circle, and therefore cos(x) = cos(-x).

The sine function is odd because it is represented by the y-coordinate of the points on the unit circle, and therefore sin(-x) = - sin(x).

The cosine function is odd because it is represented by the x-coordinate of the points on the unit circle, and
therefore cos(-x) = cos(x)

The sine function is odd because it is represented by the y-coordinate of the points on the unit circle, and therefore sin(x) = sin(-x).

Answer:

second choice is the answer or it can be

The correct answer is option B.

"A function [tex]f(x)[/tex] is an odd function when [tex]f(-x)=-f(x)[/tex] ."

For given question,

Consider points A and B on the coordinate plane as shown in following image.

Let, line segment OA makes an angle of [tex]\theta[/tex] with x-axis and ∠COB = [tex]-\theta[/tex]

So, the coordinates of points on unit circle with radius 'r' would be, A[tex](cos \theta, sin\theta)[/tex] and B[tex](cos(-\theta),sin(-\theta) )[/tex]

Let, AC = y, BC = -y and OC = m

Now, consider sine function

[tex]sin\theta=\frac{AC}{OA}\\[/tex]

⇒[tex]sin\theta=\frac{y}{r}[/tex]                        ........................(1)

And, [tex]sin(-\theta)=\frac{BC}{OB}[/tex]

⇒ [tex]sin(-\theta)=\frac{-y}{r}[/tex]

⇒ [tex]sin(-\theta)=-(\frac{y}{r})[/tex]             .......................(2)

⇒ [tex]sin(-\theta)=-sin\theta[/tex]             ......................(From (1) and (2))

Now, consider cosine function

[tex]cos\theta=\frac{OC}{OA}[/tex]

⇒ [tex]cos\theta=\frac{m}{r}[/tex]                           ...........................(3)

And, [tex]cos(-\theta)=\frac{OC}{OB}[/tex]

⇒ [tex]cos(-\theta)=\frac{m}{r}[/tex]                      ...........................(4)

⇒ [tex]cos(-\theta)=cos\theta[/tex]                   ...........................(From (3), (4))

Hence, sine function is odd because it is represented by the y-coordinate of the points on the unit circle, and therefore sin(-x) = - sin(x).

Therefore, the correct answer is option B.

Learn more about odd function here,

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