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 2

Chapter 1, Problem 2

Find the length of the arc on a circle of radius 20 feet intercepted by a 75° central angle. Express arc length in terms of 𝜋. Then round your answer to two decimal places.

Verified step by step guidance

Identify the given values: the radius \(r = 20\) feet and the central angle \(\theta = 75^\circ\).

Recall the formula for the length of an arc: \(L = r \times \theta\), where \(\theta\) must be in radians.

Convert the central angle from degrees to radians using the conversion \(\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}\), so calculate \(75^\circ \times \frac{\pi}{180}\).

Substitute the radius and the radian measure of the angle into the arc length formula: \(L = 20 \times \left(75 \times \frac{\pi}{180}\right)\).

Simplify the expression to write the arc length in terms of \(\pi\), then use a calculator to approximate the decimal value and round it to two decimal places.

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Key Concepts

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

Arc Length Formula

The arc length of a circle is the distance along the curved line forming part of the circle's circumference. It is calculated using the formula s = rθ, where r is the radius and θ is the central angle in radians. This formula connects linear and angular measurements on a circle.

Conversion Between Degrees and Radians

Angles can be measured in degrees or radians. Since the arc length formula requires the angle in radians, convert degrees to radians by multiplying by π/180. For example, 75° equals (75 × π/180) radians, which is essential for accurate calculation.

Rounding and Expressing Answers in Terms of π

Expressing the arc length in terms of π keeps the answer exact and symbolic. After finding the exact value, you can approximate the decimal value by substituting π ≈ 3.1416 and rounding to the desired decimal places, such as two decimals, for practical use.

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Related Practice

Textbook Question

A point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.

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Textbook Question

Use the unit circle shown to find the value of the trigonometric function.

cos 𝜋/6

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Textbook Question

In Exercises 1–4, a point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.

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Textbook Question

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

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

a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.

b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.

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sin 3𝜋/4

Textbook Question

In Exercises 1–8, use the Pythagorean Theorem to find the length of the missing side of each right triangle. Then find the value of each of the six trigonometric functions of θ.

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Textbook Question

In Exercises 2–4, convert each angle in degrees to radians. Express your answer as a multiple of 𝜋. 15°