In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth.20(cos 205° + i sin 205°)

Verified step by step guidance

1. Recognize that the given expression is in polar form: \( r(\cos \theta + i \sin \theta) \), where \( r = 20 \) and \( \theta = 205^\circ \).
2. Convert the polar form to rectangular form using the formula: \( x = r \cos \theta \) and \( y = r \sin \theta \).
3. Calculate \( x = 20 \cos 205^\circ \).
4. Calculate \( y = 20 \sin 205^\circ \).
5. Combine the results to express the complex number in rectangular form as \( x + yi \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, typically expressed in the form a + bi, where a is the real part and b is the imaginary part. In trigonometric form, a complex number can be represented as r(cos θ + i sin θ), where r is the magnitude and θ is the angle. Understanding how to convert between these forms is essential for solving problems involving complex numbers.

Polar to Rectangular Conversion

The conversion from polar to rectangular form involves using the relationships x = r cos θ and y = r sin θ, where (x, y) are the rectangular coordinates. This process allows us to express a complex number in the standard form a + bi. For the given complex number, we will calculate the cosine and sine of the angle and multiply by the magnitude to find the rectangular coordinates.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in determining the coordinates of points on the unit circle. These functions relate angles to ratios of sides in right triangles and are periodic in nature. In the context of complex numbers, they help in finding the real and imaginary parts when converting from polar to rectangular form, especially when dealing with angles measured in degrees.