In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 8(cos 7π/4 + i sin 7π/4)

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 $\theta = \frac{7\pi}{4}$.
2. Convert the polar form to rectangular form using the formula: $z = r \cdot (\cos \theta + i \sin \theta) = r \cdot \cos \theta + i \cdot r \cdot \sin \theta$.
3. Calculate $\cos \frac{7\pi}{4}$ and $\sin \frac{7\pi}{4}$. Note that $\frac{7\pi}{4}$ is an angle in the fourth quadrant where $\cos$ is positive and $\sin$ is negative.
4. Substitute $r = 8$, $\cos \frac{7\pi}{4}$, and $\sin \frac{7\pi}{4}$ into the rectangular form formula: $z = 8 \cdot \cos \frac{7\pi}{4} + i \cdot 8 \cdot \sin \frac{7\pi}{4}$.
5. Simplify the expression to get the rectangular form of the complex number.

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. In the given problem, the complex number is expressed in polar form, and to convert it to rectangular form, one must calculate the cosine and sine of the angle and multiply by the magnitude. This process is crucial for expressing complex numbers in a more familiar coordinate system.

Trigonometric Values of Common Angles

Knowing the trigonometric values of common angles, such as 0, π/6, π/4, π/3, and π/2, is vital for quickly evaluating sine and cosine functions. For the angle 7π/4, which corresponds to 315 degrees, the cosine and sine values can be derived from the unit circle. Familiarity with these values allows for efficient calculations when converting complex numbers from polar to rectangular form.