Cis θ

Definition

The cis function, also known as the cosine function, is a trigonometric function that represents the ratio of the adjacent side to the hypotenuse of a right triangle. It is one of the fundamental trigonometric functions, along with sine and tangent, and is essential in the study of polar form of complex numbers.

5 Must Know Facts For Your Next Test

1. The cis function is defined as $cis \theta = \cos \theta + i \sin \theta$, where $\theta$ is the angle in radians.
2. The cis function is used to represent complex numbers in polar form, where the modulus (or magnitude) is the distance from the origin and the argument (or angle) is the angle from the positive real axis.
3. The polar form of a complex number $z$ is given by $z = r \, cis \theta$, where $r$ is the modulus and $\theta$ is the argument of $z$.
4. The cis function is a convenient way to express complex numbers in polar form, as it combines the cosine and sine functions into a single expression.
5. The cis function is a powerful tool in the study of complex numbers, as it allows for easy manipulation and transformation of complex numbers, such as multiplication, division, and exponentiation.

Review Questions

• Explain the relationship between the cis function and the polar form of complex numbers.
  • The cis function, $cis \theta = \cos \theta + i \sin \theta$, is closely related to the polar form of complex numbers. In the polar form, a complex number $z$ is represented as $z = r \, cis \theta$, where $r$ is the modulus (or magnitude) of the complex number, and $\theta$ is the argument (or angle) of the complex number. The cis function allows for a compact and convenient way to express the polar form of a complex number, as it combines the cosine and sine functions into a single expression.
• Describe how the cis function can be used to perform operations on complex numbers in polar form.
  • The cis function is a powerful tool for manipulating complex numbers in polar form. For example, to multiply two complex numbers in polar form, $z_1 = r_1 \, cis \theta_1$ and $z_2 = r_2 \, cis \theta_2$, you can simply multiply their moduli and add their arguments: $z_1 \cdot z_2 = (r_1 \cdot r_2) \, cis (\theta_1 + \theta_2)$. Similarly, to divide two complex numbers in polar form, you can divide their moduli and subtract their arguments. The cis function allows for these and other operations to be performed easily on complex numbers in polar form.
• Analyze the significance of the cis function in the study of polar form of complex numbers and its broader applications in mathematics.
  • The cis function is a fundamental tool in the study of polar form of complex numbers, as it provides a compact and intuitive way to represent and manipulate these numbers. Beyond its applications in complex number theory, the cis function and its underlying trigonometric relationships have far-reaching implications across various branches of mathematics, such as in the study of periodic functions, Fourier analysis, and the representation of rotations in higher-dimensional spaces. The cis function is a unifying concept that connects complex numbers, trigonometry, and other mathematical disciplines, making it a powerful and versatile tool for mathematical analysis and problem-solving.

Related terms

Trigonometric Functions: The trigonometric functions, such as sine, cosine, and tangent, describe the relationships between the sides and angles of right triangles.

Polar Coordinates: Polar coordinates are an alternative way of specifying the location of a point in a plane, using the distance from the origin (radius) and the angle from a reference direction (angle).

Complex Numbers: Complex numbers are mathematical objects that combine a real number and an imaginary number, represented in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part.