What is the relationship between sine and cosine?

Sine and cosine are two fundamental trigonometric functions that are closely related and often used together in mathematics. The relationship between sine and cosine is defined by their definitions and properties in the context of the unit circle and right triangle trigonometry.

Unit Circle Relationship:

Sine (sin) and cosine (cos) are defined based on the unit circle, which is a circle with a radius of 1 centered at the origin of a Cartesian coordinate system.
For any point on the unit circle, the sine of the angle formed by the radius and the positive x-axis is equal to the y-coordinate of the point, and the cosine of that angle is equal to the x-coordinate of the point.
Mathematically, for an angle θ in radians:
sin(θ) = y-coordinate of the point on the unit circle
cos(θ) = x-coordinate of the point on the unit circle
Pythagorean Identity:

Another important relationship between sine and cosine is the Pythagorean identity, which relates them to each other and the number 1:
sin^2(θ) + cos^2(θ) = 1
This identity holds for any real value of θ.
Complementary Angles:

Sine and cosine are also related when dealing with complementary angles. Complementary angles are two angles whose sum is 90 degrees (π/2 radians).
If θ is an angle, then (90° - θ) is its complementary angle, and the following relationship holds:
sin(θ) = cos(90° - θ)
cos(θ) = sin(90° - θ)
Phase Shift:

Sine and cosine functions can be seen as phase-shifted versions of each other. The sine function is a cosine function with a phase shift of π/2 radians (90 degrees), and vice versa.
sin(θ) = cos(θ - π/2)
cos(θ) = sin(θ + π/2)
In summary, sine and cosine are intimately related through the unit circle, the Pythagorean identity, and the concept of complementary angles. They are two fundamental trigonometric functions that are widely used in various mathematical and scientific applications.