Ch. 1 - Angles and the Trigonometric Functions

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

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problem 12

Chapter 1, Problem 12

In Exercises 7–12, find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius, r: 1 meter Arc Length, s: 600 centimeters

Verified step by step guidance

First, ensure that the units for the radius and arc length are consistent. Since the radius is given in meters and the arc length in centimeters, convert the arc length from centimeters to meters by dividing by 100: \(s = \frac{600}{100} = 6\) meters.

Recall the formula that relates the arc length \(s\), radius \(r\), and central angle \(\theta\) in radians: \(s = r \times \theta\).

Rearrange the formula to solve for the central angle \(\theta\): \(\theta = \frac{s}{r}\).

Substitute the known values of \(s = 6\) meters and \(r = 1\) meter into the formula: \(\theta = \frac{6}{1}\).

Interpret the result as the radian measure of the central angle that intercepts the given arc length on the circle.

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Key Concepts

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

Radian Measure of an Angle

A radian is the standard unit of angular measure, defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. It provides a direct relationship between the arc length and the radius, making it essential for measuring central angles.

Relationship Between Arc Length, Radius, and Central Angle

The central angle θ in radians is calculated using the formula θ = s / r, where s is the arc length and r is the radius. This formula links linear and angular measurements, allowing conversion from arc length to angle measure in radians.

Unit Conversion

Consistent units are crucial when applying formulas. Since the radius is given in meters and the arc length in centimeters, converting one to match the other (e.g., converting 600 cm to 6 meters) ensures accurate calculation of the central angle.

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Introduction to the Unit Circle

Related Practice

Textbook Question

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of

0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋.

6 3 2 3 6 6 3 2 3 6

Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

<IMAGE>

In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.

sec 11𝜋/6

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Textbook Question

Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

<IMAGE>

tan 𝜋/3

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Textbook Question

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

In Exercises 11–18, continue to refer to the figure at the bottom of the previous page. csc 4𝜋/3

1150

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Textbook Question

In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. csc 𝜋

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Textbook Question

Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

<IMAGE>

csc 45°

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Textbook Question

In Exercises 8–12, draw each angle in standard position. -135°

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