Trigonometric Form of a Complex Number - GeeksforGeeks (2024)

In mathematics, particularly in complex number theory, the trigonometric form of the complex number plays a crucial role. This representation is not only elegant but also simplifies many operations involving complex numbers such as multiplication and division. The trigonometric form is an essential concept for students and professionals alike as it provides the geometric interpretation of complex numbers and makes complex calculations more manageable. This article aims to explore the trigonometric form of the complex numbers its derivation and its applications providing a comprehensive understanding of the topic.

What is Complex Number?

A complex number is a number that has both a real part and an imaginary part. It is of the form:

z = a + bi

Where:

• a is the real part of the complex number.
• b is the imaginary part of the complex number.
• i is the imaginary unit, defined as i = √−1​.

Components of a Complex Number

• Real Part: This is the a in the complex number a + bi. It represents the component that lies along the real axis in the complex plane.
• Imaginary Part: This is the b in the complex number a + bi. It represents the component that lies along the imaginary axis in the complex plane, multiplied by i.

What is Trigonometric Form of a Complex Number?

A complex number z can be represented in several forms. The trigonometric form also known as the polar form expresses a complex number using the trigonometric functions. It is given by:

z = r(cosθ + isinθ)

Where,

• r is the magnitude of the complex number.
• θ is the argument of the complex number.
• i is the imaginary unit where i² =−1.

Magnitude (Modulus) of Complex Number

The magnitude r of a complex number z = x + iy is given by:

[Tex]r = \sqrt{x^2 + y^2}[/Tex]

Where x and y are the real and imaginary parts of the complex number respectively.

Argument (Angle) of Complex Number

The argument [Tex]\theta [/Tex] is the angle the complex number makes with the positive real axis. It is found using:

[Tex]\theta = \tan^{-1}\left(\frac{y}{x}\right)[/Tex]

Where tan^-1 denotes the inverse tangent function.

Derivation of Trigonometric Form of Complex Number

To derive the trigonometric form from the rectangular form z = x + iy follow these steps:

Calculate the Magnitude r :

[Tex]r = \sqrt{x^2 + y^2}[/Tex]

Determine the Argument [Tex]\theta[/Tex]:

[Tex]\theta = \tan^{-1}\left(\frac{y}{x}\right)[/Tex]

Express the Complex Number in the Trigonometric Form:

[Tex]z = r \left( \cos \theta + i \sin \theta \right)[/Tex]

Solved Example: Trigonometric Form of a Complex Number

Example 1: Convert the complex number 3 + 4i to its trigonometric form.

Solution:

Find the magnitude r :

[Tex]r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5[/Tex]

Find the argument θ :

[Tex] \theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 0.93 \text{ radians}[/Tex]

Write the trigonometric form:

[Tex] 3 + 4i = 5 \left(\cos 0.93 + i \sin 0.93\right)[/Tex]

Example 2: Convert the complex number (-1 – i ) to its trigonometric form.

Solution:

Find the magnitude r :

[Tex] r = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.41[/Tex]

Find the argument θ:

[Tex] \theta = \tan^{-1}\left(\frac{-1}{-1}\right) = \tan^{-1}(1) = \frac{3\pi}{4} \text{ radians (or } 225^\circ\text{)}[/Tex]

Write the trigonometric form:

[Tex] -1 – i = \sqrt{2} \left(\cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4}\right)[/Tex]

Example 3: Find the trigonometric form of 5i.

Solution:

Find the magnitude r:

[Tex] r = \sqrt{0^2 + 5^2} = 5[/Tex]

Find the argument θ:

[Tex] \theta = \frac{\pi}{2} \text{ radians (since } 5i \text{ lies on the positive imaginary axis)}[/Tex]

Write the trigonometric form:

[Tex] 5i = 5 \left(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}\right)[/Tex]

See Also

Dot product - MATLAB dot - MathWorks United Kingdom

Example 4: Express [Tex]-2 + 2\sqrt{3}i[/Tex] in trigonometric form.

Solution:

Find the magnitude r :

[Tex] r = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4[/Tex]

Find the argument θ:

[Tex] \theta = \tan^{-1}\left(\frac{2\sqrt{3}}{-2}\right) = \tan^{-1}(-\sqrt{3}) = \frac{2\pi}{3} \text{ radians}[/Tex]

Write the trigonometric form:

[Tex] -2 + 2\sqrt{3}i = 4 \left(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}\right)[/Tex]

Example 5: Convert the complex number -4 – 4i into its trigonometric form.

Solution:

Find the magnitude r :

[Tex] r = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}[/Tex]

Find the argument θ :

[Tex] \theta = \tan^{-1}\left(\frac{-4}{-4}\right) = \frac{5\pi}{4} \text{ radians}[/Tex]

Write the trigonometric form:

[Tex] -4 – 4i = 4\sqrt{2} \left(\cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4}\right)[/Tex]

Practice Question: Trigonometric Form of a Complex Number

Q1: Convert the complex number 7 – 24i to its trigonometric form.

Q2: Express -3 + 4i in trigonometric form.

Q3: Find the trigonometric form of [Tex]2 – 2\sqrt{3}i[/Tex].

Q4: Convert the complex number 1 + i to its trigonometric form.

Q5: Write the trigonometric form of -5i.

Q6: Determine the trigonometric form of 6 + 8i.

Q7: Convert -2 – 2i to its trigonometric form.

Q8: Express 5 + 5i in trigonometric form.

Q9: Find the trigonometric form of [Tex]-1 + \sqrt{3}i[/Tex].

Q10: Convert 4 – 4i into its trigonometric form.

Conclusion

The trigonometric form of the complex number provides a powerful and intuitive way to work with complex numbers. By understanding its derivation and applications students and professionals can simplify complex number calculations and gain deeper insights into their behavior.

Read More,

• Complex Number
• Trigonometric Functions
• Domain and Range of Trigonometric Functions
• Integration of Trigonometric Functions

FAQs: Trigonometric Form of a Complex Number

What is the difference between the rectangular and trigonometric forms of a complex number?

The rectangular form is x+iy while the trigonometric form is r(cosθ+isinθ). The trigonometric form provides the geometric interpretation and simplifies the multiplication and division.

How do I convert a complex number from rectangular to trigonometric form?

Calculate the magnitude r and argument θ and then use the formula r(cosθ+isinθ).

Why is the trigonometric form useful in complex number operations?

It simplifies multiplication, division and finding the powers and roots of the complex numbers.

What is De Moivre’s Theorem?

It is a theorem that relates the powers of the complex number in trigonometric form to the trigonometric functions of the angle multiplied by the power.