Ch. 4 - Laws of Sines and Cosines; Vectors

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 4 - Laws of Sines and Cosines; Vectors

Problem 26

Chapter 4, Problem 26

In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [4(cos 50° + i sin 50°)]³

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Identify the complex number in polar form: \(4(\cos 50^\circ + i \sin 50^\circ)\), where the modulus \(r = 4\) and the argument \(\theta = 50^\circ\).

Recall DeMoivre's Theorem, which states that for a complex number in polar form, \((r(\cos \theta + i \sin \theta))^n = r^n (\cos n\theta + i \sin n\theta)\).

Apply DeMoivre's Theorem with \(n = 3\): compute the new modulus as \(r^3 = 4^3\) and the new argument as \(3 \times 50^\circ\).

Write the result in polar form: \(4^3 (\cos 150^\circ + i \sin 150^\circ)\).

Convert the polar form back to rectangular form by calculating \(4^3 \cos 150^\circ\) for the real part and \(4^3 \sin 150^\circ\) for the imaginary part.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

DeMoivre's Theorem

DeMoivre's Theorem states that for a complex number in polar form, (r(cos θ + i sin θ))^n = r^n (cos nθ + i sin nθ). It allows raising complex numbers to integer powers by multiplying the angle and raising the magnitude to the power.

Polar and Rectangular Forms of Complex Numbers

Complex numbers can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude and θ the argument, or in rectangular form as a + bi. Converting between these forms is essential for interpreting and simplifying results.

Conversion from Polar to Rectangular Form

After applying DeMoivre's Theorem, the result is in polar form. To write the answer in rectangular form, use a = r cos θ and b = r sin θ to find the real and imaginary parts, respectively, expressing the complex number as a + bi.

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Related Practice

Textbook Question

In Exercises 25–26, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.

P₁ = (-3, 0), P₂ = (-2, -2)

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Textbook Question

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.

v - u

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Textbook Question

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.

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Textbook Question

In Exercises 25–26, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.P₁ = (2, -1), P₂ = (5, -3)

Textbook Question

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.

a = 30, b = 40, A = 20°

Textbook Question

In Exercises 25–30, use Heron's formula to find the area of each triangle. Round to the nearest square unit.

a = 4 feet, b = 4 feet, c = 2 feet