Textbook Question
Determine whether each statement is true or false. See Example 4. cos 28° < sin 28° (Hint: sin 28° = cos 62°)
Ch. 2 - Acute Angles and Right Triangles
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 3, Problem 46
Solve each problem. See Examples 1–4. Distance across a Lake To find the distance RS across a lake, a surveyor lays off length RT = 53.1 m, so that angle T = 32°10' and angle S = 57°50'. Find length RS.
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Identify the triangle involved: points R, S, and T form a triangle where RT is known, and angles at T and S are given. The goal is to find the length RS.
Convert the given angles from degrees and minutes to decimal degrees for easier calculation: 32°10' becomes 32 + 10/60 degrees, and 57°50' becomes 57 + 50/60 degrees.
Calculate the third angle at point R using the triangle angle sum property: \(\angle R = 180^\circ - \angle T - \angle S\).
Use the Law of Sines to relate the sides and angles in the triangle: \(\frac{RS}{\sin(\angle T)} = \frac{RT}{\sin(\angle R)}\).
Rearrange the Law of Sines formula to solve for RS: \(RS = \frac{RT \times \sin(\angle T)}{\sin(\angle R)}\). Substitute the known values and compute the sine values to find RS.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Triangle Angle Sum Property
The sum of the interior angles in any triangle is always 180°. Knowing two angles allows you to find the third angle, which is essential for applying trigonometric laws to solve for unknown sides.
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Sum and Difference of Tangent
Law of Sines
The Law of Sines relates the lengths of sides of a triangle to the sines of their opposite angles. It states that (side/sin opposite angle) is constant for all sides, enabling calculation of unknown sides or angles when some sides and angles are known.
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4:27Intro to Law of Sines
Angle Measurement and Conversion
Angles given in degrees and minutes must be accurately interpreted or converted to decimal degrees for calculation. Understanding how to handle these measurements ensures precise use of trigonometric functions in solving the problem.
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Related Practice
Textbook Question
Evaluate each expression. See Example 4. cot² 135° - sin 30° + 4 tan 45°
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Textbook Question
Solve each right triangle. In Exercise 46, give angles to the nearest minute. In Exercises 47 and 48, label the triangle ABC as in Exercises 45 and 46. A = 39.72°, b = 38.97 m
Textbook Question
Solve each right triangle. In Exercise 46, give angles to the nearest minute. In Exercises 47 and 48, label the triangle ABC as in Exercises 45 and 46.
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Textbook Question
Determine whether each statement is true or false. See Example 4. cot 30° < tan 40°
Textbook Question
Solve each problem. See Examples 1–4. Diameter of the Sun To determine the diameter of the sun, an astronomer might sight with a transit (a device used by surveyors for measuring angles) first to one edge of the sun and then to the other, estimating that the included angle equals 32'. Assuming that the distance d from Earth to the sun is 92,919,800 mi, approximate the diameter of the sun.