Textbook Question
In Exercises 55–58, use a calculator to find the value of the acute angle θ to the nearest degree. sin θ = 0.2974
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 1, Problem 1.1.73
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: 8 feet Central Angle, θ: θ = 225°
Verified step by step guidance1
Recall the formula for the length of an arc intercepted by a central angle \( \theta \) on a circle of radius \( r \):
\[ \text{Arc length} = r \times \theta_{\text{radians}} \]
Convert the central angle from degrees to radians using the conversion factor \( \pi \text{ radians} = 180^\circ \):
\[ \theta_{\text{radians}} = 225^\circ \times \frac{\pi}{180^\circ} \]
Simplify the radian measure by reducing the fraction:
\[ \theta_{\text{radians}} = \frac{225}{180} \pi = \frac{5}{4} \pi \]
Substitute the radius \( r = 8 \) feet and the radian measure \( \theta = \frac{5}{4} \pi \) into the arc length formula:
\[ \text{Arc length} = 8 \times \frac{5}{4} \pi \]
Simplify the expression to express the arc length in terms of \( \pi \), then multiply out to find the decimal approximation and round 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 links linear distance to angular measure.
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Finding Missing Side Lengths
Conversion Between Degrees and Radians
Since the arc length formula requires the angle in radians, converting degrees to radians is essential. The conversion is done by multiplying degrees by π/180. For example, 225° equals (225 × π/180) radians, enabling correct use in trigonometric formulas.
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Expressing Answers in Terms of π and Decimal Approximation
Often, answers are first expressed symbolically in terms of π to maintain exactness, then approximated numerically for practical use. After calculating arc length with π, rounding to two decimal places provides a usable decimal value, balancing precision and clarity.
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1:37Example 1
Related Practice
Textbook Question
In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. -150°
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Textbook Question
In Exercises 55–58, use a calculator to find the value of the acute angle θ to the nearest degree. tan θ = 4.6252
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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.
14𝜋/3
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Textbook Question
If θ is an acute angle and cos θ = 1/3, find csc (𝜋/2 - θ).
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Textbook Question
In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. cos 225°
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