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 2.3.62

3장, 문제 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°

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

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

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.

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.

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.

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

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°)

671

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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)

771

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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'

608

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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.

621

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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. 2 cos 38°22' = cos 76°44'

666

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