In Exercises 15–18, write each complex number in rectangular form. If necessary, round to the nearest tenth.8(cos 60° + i sin 60°)

Verified step by step guidance

1. Recognize that the given expression is in polar form: \(r(\cos \theta + i \sin \theta)\), where \(r = 8\) and \(\theta = 60^\circ\).
2. Recall that the rectangular form of a complex number is \(a + bi\), where \(a = r \cos \theta\) and \(b = r \sin \theta\).
3. Calculate \(a\) by using \(a = 8 \cos 60^\circ\). Use the fact that \(\cos 60^\circ = \frac{1}{2}\).
4. Calculate \(b\) by using \(b = 8 \sin 60^\circ\). Use the fact that \(\sin 60^\circ = \frac{\sqrt{3}}{2}\).
5. Combine the results to express the complex number in rectangular form as \(a + bi\).

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. This process allows us to express a complex number in the standard form a + bi. For the given complex number, substituting the values of r and θ will yield the rectangular coordinates necessary for the solution.

Trigonometric Values

Trigonometric values such as cos(θ) and sin(θ) are fundamental in determining the components of a complex number in polar form. For example, cos(60°) equals 0.5 and sin(60°) equals √3/2. Knowing these values is crucial for accurately converting the complex number from its trigonometric representation to rectangular form.