Circular functions are functions that relate the angles of a right triangle to the lengths of its sides. They are also known as trigonometric functions, because they are based on the study of trigonometry. The six basic circular functions are sine, cosine, tangent, cosecant, secant, and cotangent. They can be defined using the following ratios:

• Sine: opposite/hypotenuse
• Cosine: adjacent/hypotenuse
• Tangent: opposite/adjacent
• Cosecant: hypotenuse/opposite
• Secant: hypotenuse/adjacent
• Cotangent: adjacent/opposite

Circular functions can be used to model various phenomena that involve periodic or cyclical patterns, such as sound waves, light waves, pendulums, tides, seasons, and rotations. They can also be used to solve problems involving triangles, angles, and distances. For example, circular functions can help us find the height of a building, the angle of elevation of a plane, or the length of a shadow.

Circular functions can be represented graphically using a unit circle, which is a circle with radius 1 centered at the origin of a coordinate plane. The angle formed by the positive x-axis and a ray from the origin to any point on the circle is called the standard position angle. The x-coordinate of the point is equal to the cosine of the angle, and the y-coordinate is equal to the sine of the angle. The other circular functions can be derived from these two using the ratios mentioned above. The graphs of circular functions show how their values change as the angle varies from 0 to 2π radians or 0 to 360 degrees.

One of the important properties of circular functions is that they are periodic, which means that they repeat their values after a certain interval. The smallest positive interval that makes a circular function repeat its values is called the period of the function. For example, the period of sine and cosine is 2π radians or 360 degrees, because sin(θ) = sin(θ + 2π) and cos(θ) = cos(θ + 2π) for any angle θ. The period of tangent and cotangent is π radians or 180 degrees, because tan(θ) = tan(θ + π) and cot(θ) = cot(θ + π) for any angle θ. The period of cosecant and secant is 2π radians or 360 degrees, because csc(θ) = csc(θ + 2π) and sec(θ) = sec(θ + 2π) for any angle θ.

Another important property of circular functions is that they are symmetrical, which means that they have certain relationships with their negative or complementary angles. For example, sine and cosine are even functions, which means that sin(-θ) = sin(θ) and cos(-θ) = cos(θ) for any angle θ. Tangent and cotangent are odd functions, which means that tan(-θ) = -tan(θ) and cot(-θ) = -cot(θ) for any angle θ. Cosecant and secant are also odd functions, which means that csc(-θ) = -csc(θ) and sec(-θ) = -sec(θ) for any angle θ. Moreover, sine and cosine are complementary functions, which means that sin(π/2 – θ) = cos(θ) and cos(π/2 – θ) = sin(θ) for any angle θ. Tangent and cotangent are also complementary functions, which means that tan(π/2 – θ) = cot(θ) and cot(π/2 – θ) = tan(θ) for any angle θ.

Circular functions can be extended to complex numbers using the Euler’s formula, which states that e^(i*θ) = cos(θ) + i*sin(θ), where i is the imaginary unit and e is the base of the natural logarithm. This formula allows us to define circular functions for any complex number z as follows:

• Sine: sin(z) = (e^(i*z) – e^(-i*z))/(2*i)
• Cosine: cos(z) = (e^(i*z) + e^(-i*z))/2
• Tangent: tan(z) = (e^(i*z) – e^(-i*z))/(e^(i*z) + e^(-i*z))
• Cosecant: csc(z) = 1/sin(z)
• Secant: sec(z) = 1/cos(z)
• Cotangent: cot(z) = 1/tan(z)