Write the Complex Number in Trigonometric (Polar) Form 4sqrt(3) + 4i

TL;DR

Learn how to write a complex number in trigonometric form by finding the angle and using the formula r(cos θ + i sin θ).

Transcript

write the complex number in trigonometric form so the first thing i like to do in these problems when we're asked to write in trig form is to do a rough sketch of the complex number so there's the imaginary plane and this is the real part and this here is the called the real axis and this is called the imaginary axis and so they're both positive so... Read More

Key Insights

• 😘 Trigonometric form of a complex number is r(cos θ + i sin θ).
• #️⃣ The magnitude of a complex number (r) can be calculated using the formula r = √(x^2 + y^2).
• ☺️ θ can be found using the tangent formula (θ = tan^(-1)(y/x)) or the system of equations method.
• 💁 The trigonometric form can also be written as 8 cis(pi/6) using the cis notation.
• 🎭 Understanding trigonometric form helps in performing operations on complex numbers.
• 🥳 The real and imaginary parts of a complex number must be equal when using the trigonometric form.
• 💁 Trigonometric form provides a visual representation of a complex number on the imaginary plane.

Questions & Answers

Q: How do I write a complex number in trigonometric form?

To write a complex number in trigonometric form, first sketch it on the imaginary plane, calculate the value of r, find the value of θ using either the tangent formula or the system of equations method, and then plug the values into the trigonometric form formula.

Q: What is the formula for calculating r?

The formula for calculating r is r = √(x^2 + y^2), where x and y are the real and imaginary parts of the complex number.

Q: Why is θ often found using the tangent formula?

Many people use the tangent formula (θ = tan^(-1)(y/x)) to find the value of θ because it is convenient and eliminates the need for memorizing values. However, the system of equations method can also be used.

Q: What is the trigonometric form of a complex number?

The trigonometric form of a complex number is r(cos θ + i sin θ), where r is the magnitude of the complex number, and θ is the angle it makes with the positive real axis.

Summary & Key Takeaways

• To write a complex number in trigonometric form, start by sketching the number on the imaginary plane and identifying the real and imaginary parts.

• Calculate the value of r using the formula r = √(x^2 + y^2), where x and y are the real and imaginary parts of the complex number.

• Use the tangent of θ formula (y/x) or the system of equations method to find the value of θ.

• Plug the values of r and θ back into the trigonometric form formula to obtain the complex number in trigonometric form.