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 1.1.71

Chapter 1, Problem 1.1.71

In Exercises 71–74, find the length of the arc on a circle of radius r intercepted by a central angle θ. Express arc length in terms of 𝜋. Then round your answer to two decimal places. Radius, r: 12 inches Central Angle, θ: θ = 45°

Verified step by step guidance

Recall the formula for the length of an arc \(s\) intercepted by a central angle \(\theta\) on a circle of radius \(r\): \(s = r \times \theta_{\text{radians}}\)

Since the central angle \(\theta\) is given in degrees, convert it to radians using the conversion factor: \(\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}\)

Substitute the given values into the conversion formula: \(\theta_{\text{radians}} = 45 \times \frac{\pi}{180}\)

Now substitute the radius \(r = 12\) inches and the radian measure of \(\theta\) into the arc length formula: \(s = 12 \times \left(45 \times \frac{\pi}{180}\right)\)

Simplify the expression to write the arc length in terms of \(\pi\), then calculate the decimal approximation and round it to two decimal places.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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 between two points on the circle. It is calculated using the formula s = rθ, where r is the radius and θ is the central angle in radians. This formula directly relates the angle and radius to the length of the arc.

Conversion Between Degrees and Radians

Since the arc length formula requires the central angle in radians, it is essential to convert degrees to radians. The conversion is done by multiplying degrees by π/180. For example, 45° equals 45 × (π/180) = π/4 radians.

Expressing Answers in Terms of π and Decimal Approximation

After calculating the arc length in terms of π, it is often useful to provide a decimal approximation for practical use. This involves substituting π ≈ 3.1416 and rounding the result to the desired decimal places, such as two decimals, to balance precision and readability.

Recommended video:

1:37

Example 1

Related Practice

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. (-4, 3)

542

views

Textbook Question

In Exercises 63–68, find the exact value of each expression. Do not use a calculator. cos 12° sin 78° + cos 78° sin 12°

591

views

Textbook Question

In Exercises 63–68, find the exact value of each expression. Do not use a calculator. tan(𝜋/3)/2 - 1/sec(𝜋/6)

612

views

Textbook Question

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

120°

635

views

Textbook Question

In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. 395°

679

views

Textbook Question

In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. cot(7𝜋/4)

657

views