Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

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

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problem 35

Chapter 5, Problem 35

In Exercises 29–36, simplify and write the result in standard form. ____________ √1² − 4 ⋅ 0.5 ⋅ 5

Verified step by step guidance

Identify the expression under the square root, which is \(1^2 - 4 \cdot 0.5 \cdot 5\). This is a part of the quadratic formula's discriminant, but here we just need to simplify it.

Calculate each part inside the square root separately: first, square 1 to get \(1^2 = 1\).

Multiply the constants: \(4 \cdot 0.5 = 2\), then multiply by 5 to get \(2 \cdot 5 = 10\).

Substitute these values back into the expression under the square root: \(1 - 10\).

Simplify the expression under the square root to get \(-9\), then recognize that the square root of a negative number involves imaginary numbers, so write it as \(\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1}\).

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Key Concepts

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

Order of Operations

The order of operations dictates the sequence in which mathematical operations are performed to ensure consistent results. It follows the PEMDAS/BODMAS rules: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Applying this correctly is essential when simplifying expressions involving roots and arithmetic.

Square Root and Radicals

The square root of a number is a value that, when multiplied by itself, gives the original number. Simplifying square roots often involves evaluating the expression inside the radical first. Understanding how to handle radicals and simplify them is crucial for expressing results in standard form.

Standard Form of a Number

Standard form refers to expressing a number in a simplified, conventional way, often as a decimal or a simplified radical without complex components. In trigonometry and algebra, writing answers in standard form ensures clarity and uniformity, making it easier to interpret and use the results.

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In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex cube roots of −1

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In Exercises 35–44, test for symmetry and then graph each polar equation. r = cos θ/2

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In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 20(cos 205° + i sin 205°)

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In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (4, 90°)

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In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞.

x = 2 + 4 cos t, y = −1 + 3 sin t; 0 ≤ t ≤ π

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In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex fifth roots of −1 − i