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 53
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.
Verified step by step guidance1
Identify the sides of the right triangle relative to the given angle of 39°. Here, side QR (51 m) is opposite angle P, and side RP is adjacent to angle P.
Determine which side length you need to find. For example, if you want to find side QP (the hypotenuse), use the sine or cosine function based on the given angle and known side.
Set up the trigonometric ratio using the angle and known side. For example, if finding the hypotenuse QP, use the cosine function: \(\cos(39^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{51}{QP}\).
Rearrange the equation to solve for the unknown side. For the hypotenuse, multiply both sides by QP and then divide both sides by \(\cos(39^\circ)\) to isolate QP: \(QP = \frac{51}{\cos(39^\circ)}\).
Calculate the value using a calculator (make sure it is in degree mode), then round the answer to the nearest whole number as required.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Right Triangle Properties
A right triangle has one angle of 90 degrees, and the other two angles sum to 90 degrees. The sides are related through trigonometric ratios, and the side opposite the right angle is the hypotenuse, the longest side.
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30-60-90 Triangles
Trigonometric Ratios (Sine, Cosine, Tangent)
Sine, cosine, and tangent relate the angles of a right triangle to the ratios of its sides. For an angle θ, sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, and tangent = opposite/adjacent. These ratios help find unknown side lengths.
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5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Using Given Angle and Side to Find Unknown Sides
Given one angle (other than 90°) and one side length, you can use trigonometric ratios to find the lengths of the other sides. Identify which side corresponds to opposite, adjacent, or hypotenuse relative to the given angle to apply the correct ratio.
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4:18Finding Missing Side Lengths
Related Practice
Textbook Question
In Exercises 44–48, find the reference angle for each angle.
- 11𝜋/3
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Textbook Question
Use reference angles to find the exact value of each expression. Do not use a calculator. cot 19𝜋/6
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Textbook Question
Use reference angles to find the exact value of each expression. Do not use a calculator. sec 495°
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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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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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