Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161당신이 사용하는 게 아니라요?교과서 변경

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

문제 2.3.56

3장, 문제 2.3.56

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'

검증된 단계별 안내

First, convert the given angles from degrees and minutes to decimal degrees. For example, 38°22' can be converted by calculating \(38 + \frac{22}{60}\) degrees.

Calculate the left side of the equation by evaluating \(2 \times \cos(38.3667^\circ)\) using a calculator, where 38.3667° is the decimal equivalent of 38°22'.

Calculate the right side of the equation by evaluating \(\cos(76.7333^\circ)\), where 76.7333° is the decimal equivalent of 76°44'.

Compare the two results obtained from the left and right sides. If they are equal or differ only in the last decimal place due to rounding, the statement is true; otherwise, it is false.

Optionally, recall the double-angle identity for cosine: \(2 \cos \theta = \cos(2\theta)\) is not generally true, so this can help understand why the equality might or might not hold.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Trigonometric Functions and Their Properties

Cosine is a fundamental trigonometric function that relates an angle in a right triangle to the ratio of the adjacent side over the hypotenuse. Understanding how cosine values change with different angles is essential for evaluating and comparing expressions involving cosine.

Angle Measurement and Conversion

Angles can be expressed in degrees, minutes, and seconds, where 1 degree = 60 minutes and 1 minute = 60 seconds. Accurate conversion and interpretation of these units are crucial for precise calculations and comparisons in trigonometry.

Use of Calculators and Rounding Errors

Calculators approximate trigonometric values, which can lead to small rounding errors, especially in decimal places. Recognizing that minor differences may occur due to rounding helps in correctly interpreting the truth value of trigonometric statements.

관련 실천

교과서 질문

Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.

tan θ = 6.4358841

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교과서 질문

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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교과서 질문

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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교과서 질문

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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교과서 질문

Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.

csc θ = 1.3861147

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교과서 질문

(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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