# What are the definitions of the sine and cosine functions?

## What are the definitions of the sine and cosine functions?

Sine and cosine — a.k.a., sin(θ) and cos(θ) — are functions revealing the shape of a right triangle. Looking out from a vertex with angle θ, sin(θ) is the ratio of the opposite side to the hypotenuse , while cos(θ) is the ratio of the adjacent side to the hypotenuse .

### What is the sine and cosine graph?

The basic sine and cosine functions have a period of 2π. The function sin x is odd, so its graph is symmetric about the origin. The function cos x is even, so its graph is symmetric about the y-axis. The graph of a sinusoidal function has the same general shape as a sine or cosine function.

**What is complex sin?**

Recall that the complex sine function is defined, for z ∈ C, as: sinz def = eiz − e−iz 2i . The goal of this section is to show that this extension of the usual sine function of calculus to the complex plane does not add any new zeros.

**What is cosine of a complex number?**

These formulas are often used as definitions of the trigonometric functions for complex numbers. cosx=eix+e−ix2,sinx=eix−e−ix2i. Substituting the quantity ix in place of the variable x, we can produce formulas for pure imaginary numbers. Notice that the results are related to the hyperbolic functions.

## How is sine defined?

Definition of sine 1 : the trigonometric function that for an acute angle is the ratio between the leg opposite the angle when it is considered part of a right triangle and the hypotenuse.

### Why is it important to have both the right triangle definitions of sine and cosine?

The right triangle definitions of sine and cosine only apply to acute angles, so a more complete definition is needed. The point where the terminal side intersects the unit circle (x, y) is the basis for this definition.

**What is a cosine graph?**

Cos Graph. The cosine graph or the cos graph is an up-down graph just like the sine graph. The only difference between the sine graph and the cos graph is that sine graph starts from 0 while the cos graph starts from 90 (or π/2). The cos graph given below starts from 1 and falls till -1 and then starts rising again.

**What is the graph of sine function?**

To graph the sine function, we mark the angle along the horizontal x axis, and for each angle, we put the sine of that angle on the vertical y-axis. The result, as seen above, is a smooth curve that varies from +1 to -1. Curves that follow this shape are called ‘sinusoidal’ after the name of the sine function.

## What is Euler’s formula in complex numbers?

Euler’s formula is the statement that e^(ix) = cos(x) + i sin(x). When x = π, we get Euler’s identity, e^(iπ) = -1, or e^(iπ) + 1 = 0.

### Is Cos the real or imaginary?

cos denotes the cosine function (real and complex) sin denotes the real sine function. sinh denotes the hyperbolic sine function. cosh denotes the hyperbolic cosine function.

**How is cosine defined?**

Definition of cosine 1 : a trigonometric function that for an acute angle is the ratio between the leg adjacent to the angle when it is considered part of a right triangle and the hypotenuse. 2 : a trigonometric function cos θ that for all real numbers θ is given by the sum of the alternating series 1−x22! +x44!