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.

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.

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.