Can you take the cosine of an imaginary number?

These formulas are often used as definitions of the trigonometric functions for complex numbers. Substituting the quantity ix in place of the variable x, we can produce formulas for pure imaginary numbers. We get cosi=cos0cosh1−isin0sinh1=cosh1=e1+e−12≈1.5431.

Why does Cos x )= 3 have no solution?

The range of cosine is −1≤y≤1 – 1 ≤ y ≤ 1 . Since −3 does not fall in this range, there is no solution.

Are trigonometric functions defined for complex numbers?

The trigonometric functions can be defined for complex variables as well as real ones. One way is to use the power series for sin(x) and cos(x), which are convergent for all real and complex numbers. An easier procedure, however, is to use the identities from the previous section: cos(i x) = cosh(x)

How do you find the COS of a complex number?

Let a and b be real numbers. Let i be the imaginary unit. Then: cos(a+bi)=cosacoshb−isinasinhb.

What is Euler’s identity used for?

We use Euler’s identity to show how the constants e, pi, and i are related. Euler’s formula is used to help us calculate e to an imaginary power. Let’s see how we go about calculating a sample number.

Why does Cos have no solution?

This strange truth results from the fact that the trigonometric functions are periodic, repeating every 360 degrees or 2Π radians. Because the set of values from 0 to 2Π contains the domain for all six trigonometric functions, if there is no solution to an equation between these bounds, then no solution exists.

Is Cos 3 x even or odd?

Therefore, cos3(−x)=cos(−x)cos(−x)cos(−x)=cosxcosxcosx=cos3x (i.e. cos3x must be even function). And similarly, since sin(−x)=−sinx, sin3x must be odd function.

How do you find the value of imaginary numbers?

To calculate the value of an imaginary number, we use the notation iota or “i”. The square root of a negative number gives us an imaginary number. We use this value of i in understanding the concepts of complex numbers. Here √-1 is the imaginary part.

What is an imaginary solution to a quadratic equation?

Imaginary or complex roots will occur when the value under the radical portion of the quadratic formula is negative. Notice that the value under the radical portion is represented by “b2 – 4ac”. So, if b2 – 4ac is a negative value, the quadratic equation is going to have complex conjugate roots (containing “i “s).

Is for complex number z?

We often use the variable z=a+bi to represent a complex number. The number a is called the real part of z: Re z while b is called the imaginary part of z: Im z. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

What is the maximum and minimum value of cosx?

Cosx cannot be equal to 2. Hence for no value of x the above condition can be satisfied. The maximum value of cosx is 1 and minimum value of cosx is -1. We need to use Euler’s formula for this question.

What is Euler’s formula for cos 1 sin 2?

The central mathematical fact that we are interested in here is generally called Euler’s formula”, and written ei= cos+ isin Using equations 2 the real and imaginary parts of this formula are cos= 1 2 (ei+ e i) sin= 1 2i (ei e i) (which, if you are familiar with hyperbolic functions, explains the name of the hyperbolic cosine and sine).

What is ei = Cos + Isin?

3 Euler’s formula The central mathematical fact that we are interested in here is generally called \\Euler’s formula”, and written ei = cos + isin Using equations 2 the real and imaginary parts of this formula are cos = 1 2 (ei + e i ) sin = 1 2i (ei e i ) (which, if you are familiar with hyperbolic functions, explains the name of the

What is the radii of convergence of cosine and sine?

complex sine and cosine We define for all complex values of z: Because these series converge for all real values of z, their radii of convergence are ∞, and therefore they converge for all complex values of z (by a known of Abel; cf. the entry power series), too.