Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 1 - Functions

Problem 1.3.65

Chapter 1, Problem 1.3.65

Refer to the given figure. Write the radius r of the circle in terms of α and θ.

Verified step by step guidance

Step 1: Begin by understanding the relationship between the angle α, angle θ, and the radius r of the circle. Typically, these angles are part of a geometric configuration involving circles, such as sectors or segments.

Step 2: Consider the geometric properties of the circle. If α and θ are angles subtended by arcs or sectors, they might relate to the radius through trigonometric identities or geometric formulas.

Step 3: Use trigonometric identities or geometric relationships to express r in terms of α and θ. For example, if α and θ are angles in a right triangle formed by the radius, you might use sine, cosine, or tangent functions.

Step 4: If the figure involves a sector of the circle, recall that the arc length or area of the sector can be used to relate the radius to the angles. The formula for arc length is L = rθ, where θ is in radians.

Step 5: Combine the relationships and formulas derived from the figure to express the radius r solely in terms of α and θ. Ensure that any assumptions made about the figure are consistent with the given angles and their geometric context.

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Key Concepts

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

Circle Geometry

Understanding the properties of circles is essential for solving problems involving their dimensions. The radius is a key feature, defined as the distance from the center of the circle to any point on its circumference. In terms of angles, the relationship between the radius and angles like α and θ can often be explored using trigonometric functions.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent relate the angles of a triangle to the ratios of its sides. In the context of a circle, these functions can help express the radius in terms of angles. For example, if α and θ represent angles in a right triangle inscribed in the circle, the radius can be derived using these functions.

Coordinate Systems

Coordinate systems, particularly polar coordinates, are often used to describe points in relation to a circle. In polar coordinates, a point is defined by its distance from the origin (the radius) and the angle from the positive x-axis. Understanding how to convert between polar and Cartesian coordinates is crucial for expressing the radius in terms of angles like α and θ.

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Textbook Question

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

g(x) = x/(x² − 1)

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Textbook Question

Finding a Viewing Window

In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.

y = x + (1/10) sin 30x

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Textbook Question

Shifting Graphs

The accompanying figure shows the graph of y = −x² shifted to two new positions. Write equations for the new graphs.

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Textbook Question

Using the Half-Angle Formulas

Find the function values in Exercises 47–50.