Textbook Question
Find the area of the sector of a circle of radius r formed by a central angle θ. Express area in terms of π. Then round your answer to two decimal places. Radius, r: 4 inches Central Angle, θ: θ = 240°
Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 1, Problem 49
In Exercises 49–54, find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round answers to the nearest whole number.
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Identify the sides of the right triangle relative to the given angle of 21° at vertex P. The side PR = 413 m is adjacent to angle P, and side QR is opposite to angle P. The hypotenuse is PQ.
To find the length of the side opposite to angle P (which is QR), use the sine function, since sine relates the opposite side to the hypotenuse: \(\sin(21^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{QR}{PQ}\).
To find the length of the hypotenuse PQ, use the cosine function, since cosine relates the adjacent side to the hypotenuse: \(\cos(21^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{PR}{PQ} = \frac{413}{PQ}\).
Rearrange the cosine equation to solve for the hypotenuse PQ: \(PQ = \frac{413}{\cos(21^\circ)}\).
Once you find PQ, substitute it back into the sine equation to solve for QR: \(QR = PQ \times \sin(21^\circ)\). This will give you the length of the side opposite the 21° angle.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Right Triangle Trigonometry
Right triangle trigonometry involves relationships between the angles and sides of a right triangle. The primary trigonometric ratios—sine, cosine, and tangent—relate an angle to the ratios of two sides. These ratios are essential for finding unknown side lengths or angles when some measurements are known.
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Trigonometric Ratios (Sine, Cosine, Tangent)
Sine, cosine, and tangent are ratios defined for an acute angle in a right triangle: sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent. These ratios allow calculation of unknown sides or angles when one side length and one angle are known.
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Using Given Angle and Side to Find Unknown Side
Given an angle and one side length in a right triangle, you can use trigonometric ratios to find the length of another side. For example, with angle 21° and adjacent side 413 m, cosine can find the hypotenuse, or tangent can find the opposite side. Rounding to the nearest whole number is often required for practical answers.
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Related Practice
Textbook Question
In Exercises 44–48, find the reference angle for each angle.
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Textbook Question
In Exercises 39–48, use a calculator to find the value of the trigonometric function to four decimal places.
cot 𝜋/12
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Textbook Question
In Exercises 49–54, find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round answers to the nearest whole number.
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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.
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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.
16𝜋/3
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