Visualization: 

Consider a unit circle (a circle of radius 1). Consider a line drawn from the origin O, that makes an angle θ with the X-axis in the anti-clockwise direction. This line intersects the circle at a point P. The length of the projection of the line segment OP on the Y-axis is equal to the Sine of θ.

Relationship between Sine and Cosine:

Cosine has a serendipitous co-existence with Sine. The length of projection of the same line segment OP on the X-axis is equal to the Cosine of θ

The basic trigonometric identity:

Just using Pythagoras theorem, we see that sin(θ)^2 + cos(θ)^2 = 1 for all angles θ.

Value of Sines and Cosines of multiples of 90 degrees:

1. The Sine of 0 degrees is 0, since a line coinciding with the X-axis has no projection on the Y-axis.
2. The Cosine of 0 degrees is 1, since a line drawn within a unit circle and coinciding with the X-axis has a projection of length 1 on the X-axis.
3. The Sine of 90 degrees is 1, since a line drawn within a unit circle and coinciding with the Y-axis has a projection of length 1 on the Y-axis.
4. The Cosine of 90 degrees is 0, since a line coinciding with the Y-axis has no projection on the X-axis.
5. The same logic holds good for 180 degrees and 270 degrees.
6. This visualization also allows us to extend the definition of trigonometric functions beyond 90 degrees, since we are no longer restricted to a triangle.

Signs of Sine and Cosine functions:

1. Any angle from 0 to 180 degrees has a positive Sine, since the length of projection on the Y-axis is measured along the positive Y-direction.
2. Sine is negative for the rest of the angles because the length of projection on the Y-axis is measured along the negative Y-direction.
3. Any angle from 0 to 90 degrees and 270 to 360 degrees has a positive Cosine, since the length of projection on the X-axis is measured along the positive X-direction.
4. Cosine is negative for rest of the angles because the length of projection on the X-axis is measured along the negative X-direction.

Sine is an odd function:         $$\sin (-\theta )=-\sin \left(\theta \right)$$

1. Consider an angle θ, having a magnitude between 0 and 90 degrees.
2. The Sine of +θ is always positive since the projection on the Y-axis is always along the positive Y-direction.
3. The Sine of -θ is always negative since the projection on the Y-axis is always along the negative Y-direction.
4. Hence, sin(-θ) = -sin(θ): Sine is an odd function.
5. The same also holds good for an angle θ having a magnitude between 90 and 180 degrees: Sin(θ) is always positive whereas sin(-θ) is always negative.

Cosine is an even function:         $$\cos (-\theta )=\cos \left(\theta \right)$$

1. Consider an angle θ, having a magnitude between 0 and 90 degrees.
2. The Cosine of +θ is always positive since the projection on the X-axis is always along the positive X-direction.
3. The Cosine of -θ is always positive since the projection on the X-axis is always along the positive X-direction.
4. Hence, cos(-θ) = cos(θ): Cosine is even odd function.
5. The same also holds good for an angle θ having a magnitude between 90 and 180 degrees: Cos(θ) and cos(-θ) are both negative.