Textbook Question
Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1.
1/csc(90°-51°)
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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 2.3.62
Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error. cos(30° + 20°) = cos 30° + cos 20°
Verified step by step guidance1
Recall the cosine addition formula: \(\cos(A + B) = \cos A \cos B - \sin A \sin B\).
Apply the formula to the left side of the equation: \(\cos(30^\circ + 20^\circ) = \cos 30^\circ \cos 20^\circ - \sin 30^\circ \sin 20^\circ\).
Calculate the right side of the equation as given: \(\cos 30^\circ + \cos 20^\circ\).
Use a calculator to find the numerical values of both sides: compute \(\cos 30^\circ \cos 20^\circ - \sin 30^\circ \sin 20^\circ\) and \(\cos 30^\circ + \cos 20^\circ\) separately.
Compare the two results to determine if the original statement is true or false, keeping in mind that small differences in the last decimal place may be due to rounding errors.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cosine of a Sum Formula
The cosine of the sum of two angles is given by cos(A + B) = cos A cos B - sin A sin B. This identity is fundamental for evaluating expressions involving the cosine of angle sums and helps verify if an equation involving cos(30° + 20°) is true.
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Verifying Identities with Sum and Difference Formulas
Properties of Trigonometric Functions
Trigonometric functions like cosine are nonlinear and do not distribute over addition, meaning cos(A + B) is not equal to cos A + cos B. Understanding this property is essential to avoid common misconceptions when comparing trigonometric expressions.
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Rounding Errors in Calculator Computations
Calculators approximate trigonometric values, which can cause minor differences in the last decimal places. Recognizing that small discrepancies may arise from rounding helps in correctly interpreting the truth value of trigonometric statements.
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Related Practice
Textbook Question
Solve each problem. See Examples 1 and 2. Distance between Two Ships A ship leaves its home port and sails on a bearing of S 61°50'. Another ship leaves the same port at the same time and sails on a bearing of N 28°10'E. If the first ship sails at 24.0 mph and the second sails at 28.0 mph, find the distance between the two ships after 4 hr.
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Textbook Question
Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. Suppose that an observer for a radar station is located at the origin of a coordinate system. Find the bearing of an airplane located at each point. Express the bearing using both methods. (-3, -3)
Textbook Question
Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1.
cos 41° 24'
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Textbook Question
Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error. tan² 72°25' + 1 = sec² 72°25'
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Textbook Question
Solve each problem. See Examples 3 and 4. Height of an Antenna A scanner antenna is on top of the center of a house. The angle of elevation from a point 28.0 m from the center of the house to the top of the antenna is 27°10', and the angle of elevation to the bottom of the antenna is 18°10'. Find the height of the antenna.