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(90°-3.69°)
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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 2.3.12
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'
Verified step by step guidance1
First, convert the angle given in degrees and minutes to a decimal degree format. Recall that 1 minute (') is equal to \( \frac{1}{60} \) degrees. So, convert 24' to degrees by calculating \( 24 \times \frac{1}{60} \).
Add the decimal degree value from the minutes to the whole degrees to get the total angle in decimal degrees: \( 41 + \text{(decimal from minutes)} \).
Use the cosine function on your calculator with the angle in decimal degrees. Make sure your calculator is set to degree mode, not radians.
Calculate \( \cos(41.4°) \) (or the exact decimal degree you found) using the calculator to get the approximate value.
Round the result to six decimal places as requested.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Angle Measurement in Degrees and Minutes
Angles can be expressed in degrees, minutes, and seconds, where 1 degree equals 60 minutes. To use a calculator, angles given in degrees and minutes must be converted to decimal degrees by dividing the minutes by 60 and adding to the degrees.
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Reference Angles on the Unit Circle
Using the Cosine Function
The cosine function relates an angle in a right triangle to the ratio of the adjacent side over the hypotenuse. Calculators compute cosine values for angles in degrees or radians, so the angle must be correctly input in the calculator's mode.
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Rounding and Decimal Precision
When approximating values using a calculator, it is important to round the result to the specified number of decimal places. Here, answers should be rounded to six decimal places to ensure accuracy and consistency.
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Related Practice
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)
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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. cos(30° + 20°) = cos 30° + cos 20°
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Textbook Question
(Modeling) Length of a Sag Curve When a highway goes downhill and then uphill, it has a sag curve. Sag curves are designed so that at night, headlights shine sufficiently far down the road to allow a safe stopping distance. See the figure. S and L are in feet. The minimum length L of a sag curve is determined by the height h of the car's headlights above the pavement, the downhill grade θ₁ < 0°, the uphill grade θ₂ > 0°, and the safe stopping distance S for a given speed limit. In addition, L is dependent on the vertical alignment of the headlights. Headlights are usually pointed upward at a slight angle α above the horizontal of the car. Using these quantities, for a 55 mph speed limit, L can be modeled by the formula (θ₂ - θ₁)S² L = ————————— , 200(h + S tan α) where S < L. (Data from Mannering, F., and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.) Compute length L, to the nearest foot, if h = 1.9 ft, α = 0.9°, θ₁ = -3°, θ₂ = 4°, and S = 336 ft.
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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. 2 cos 38°22' = cos 76°44'
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Textbook Question
Find two angles in the interval [0°, 360°) that satisfy each of the following. Round answers to the nearest degree. tan θ = 0.70020753
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