In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 6(cos 30° + i sin 30°)

Verified step by step guidance

1. Recognize that the given expression is in polar form, represented as \( r(\cos \theta + i \sin \theta) \), where \( r = 6 \) and \( \theta = 30^\circ \).
2. Use the formula for converting from polar to rectangular form: \( z = r \cos \theta + i r \sin \theta \).
3. Substitute the values of \( r \) and \( \theta \) into the formula: \( z = 6 \cos 30^\circ + i \cdot 6 \sin 30^\circ \).
4. Calculate \( \cos 30^\circ \) and \( \sin 30^\circ \) using known trigonometric values: \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) and \( \sin 30^\circ = \frac{1}{2} \).
5. Substitute these trigonometric values back into the expression to find the rectangular form: \( z = 6 \cdot \frac{\sqrt{3}}{2} + i \cdot 6 \cdot \frac{1}{2} \).

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 coefficient of the imaginary unit i. 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. For a complex number in polar form r(cos θ + i sin θ), this means calculating the real part as r cos θ and the imaginary part as r sin θ. This conversion is crucial for expressing complex numbers in a more familiar coordinate system.

Trigonometric Values

Trigonometric values such as sin and cos are fundamental in determining the coordinates of points on the unit circle. For example, cos 30° and sin 30° correspond to specific values: cos 30° = √3/2 and sin 30° = 1/2. Knowing these values allows for accurate calculations when converting complex numbers from polar to rectangular form, ensuring precision in the final result.