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 7

Chapter 1, Problem 7

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: 10 inches Arc Length, s: 40 inches

Verified step by step guidance

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

Identify the given values from the problem: radius \(r = 10\) inches and arc length \(s = 40\) inches.

Substitute the known values into the formula: \[ 40 = 10 \times \theta \]

Solve for the central angle \(\theta\) by dividing both sides of the equation by the radius \(r\): \[ \theta = \frac{40}{10} \]

Simplify the fraction to express \(\theta\) in radians, which will give the radian measure of the central angle.

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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 a unit of angular measure based on the radius of a circle. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius. It provides a natural way to relate angles to arc lengths without converting to degrees.

Relationship Between Arc Length, Radius, and Central Angle

The central angle θ (in radians) of a circle is related to the arc length s and radius r by the formula θ = s / r. This formula allows you to find the angle when the arc length and radius are known, making it essential for solving problems involving circular arcs.

Units and Conversion in Trigonometry

Understanding units is crucial; arc length is measured in linear units (e.g., inches), radius in the same units, and the central angle in radians (a dimensionless measure). Ensuring consistent units and interpreting the result in radians is key to correctly solving and applying trigonometric problems.

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Related Practice

Textbook Question

In Exercises 8–12, draw each angle in standard position. 5𝜋 6

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Textbook Question

In Exercises 5–7, convert each angle in radians to degrees. - 5𝜋 6

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Textbook Question

The unit circle has been divided into twelve equal arcs, corresponding to t-values of

0, 𝜋/6, 𝜋/3, 𝜋/2, 2𝜋/3, 5𝜋/6, 𝜋, 7𝜋/6, 4𝜋/3, 3𝜋/2, 5𝜋/3, 11𝜋/6, and 2𝜋

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.

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cos 5𝜋/6

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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.

cos 2𝜋/3

Textbook Question

In Exercises 1–8, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ. (-1, -3)

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Textbook Question

In Exercises 1–8, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ. (5, -5)

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