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)

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Recognize that the given complex number is in polar (trigonometric) form: $r(\cos \theta + i \sin \theta)$, where $r = 8$ and $\theta = \frac{7\pi}{4}$.

Recall that to convert from polar form to rectangular form, use the formulas: $x = r \cos \theta$ and $y = r \sin \theta$, where $x$ is the real part and $y$ is the imaginary part.

Calculate the real part: $x = 8 \times \cos \left( \frac{7\pi}{4} \right)$.

Calculate the imaginary part: $y = 8 \times \sin \left( \frac{7\pi}{4} \right)$.

Write the rectangular form as $x + yi$, substituting the values found for $x$ and $y$. If necessary, round the values to the nearest tenth.

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Polar Form of Complex Numbers

A complex number can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude (distance from the origin) and θ is the argument (angle with the positive real axis). This form is useful for multiplication, division, and finding powers or roots of complex numbers.

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Complex Numbers In Polar Form

Conversion from Polar to Rectangular Form

To convert a complex number from polar to rectangular form, use the formulas x = r cos θ and y = r sin θ, where x is the real part and y is the imaginary part. The rectangular form is written as x + iy, which is often easier to interpret and use in algebraic operations.

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Converting Complex Numbers from Polar to Rectangular Form

Evaluating Trigonometric Functions at Specific Angles

Understanding the values of cosine and sine at common angles, such as 7π/4, is essential. For 7π/4 (or 315°), cos(7π/4) = √2/2 and sin(7π/4) = -√2/2. These values help accurately convert the complex number into its rectangular form.

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