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 2.3.48

3장, 문제 2.3.48

Use a calculator to evaluate each expression. 2 sin 25°13' cos 25°13' - sin 50°26'

검증된 단계별 안내

First, recognize that the angles are given in degrees and minutes. Convert the angle 25°13' into decimal degrees if needed, or use the angle directly in your calculator by inputting degrees and minutes appropriately.

Recall the double-angle identity for sine: \(\sin(2\theta) = 2 \sin \theta \cos \theta\). Notice that the expression \(2 \sin 25°13' \cos 25°13'\) matches the left side of this identity with \(\theta = 25°13'\).

Rewrite the expression \(2 \sin 25°13' \cos 25°13'\) as \(\sin(2 \times 25°13')\), which simplifies to \(\sin 50°26'\).

Substitute this back into the original expression to get \(\sin 50°26' - \sin 50°26'\).

Since the two terms are the same, their difference is zero. This shows the expression simplifies to zero without needing further calculation.

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영상 길이:

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주요 개념

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

Trigonometric Functions and Their Values

Sine and cosine are fundamental trigonometric functions that relate angles to ratios of sides in a right triangle. Understanding how to evaluate these functions for given angles, including those expressed in degrees and minutes, is essential for solving the expression.

Angle Conversion and Notation

Angles given in degrees and minutes (e.g., 25°13') must be accurately interpreted or converted to decimal degrees for calculator input. One minute equals 1/60 of a degree, so converting ensures precise evaluation of trigonometric functions.

Trigonometric Identities

The expression involves terms like 2 sin A cos A and sin 2A, which are connected by the double-angle identity: sin 2A = 2 sin A cos A. Recognizing and applying this identity simplifies the expression and aids in verifying the result.

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. cot 183° 48'

650

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교과서 질문

CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.

Column I: 1.

csc⁻¹ 4

Column II:

A. 88.09084757°

B. 63.25631605°

C. 1.909152433°

D. 17.45760312°

E. 0.2867453858

F. 1.962610506

G. 14.47751219°

H. 1.015426612

I. 1.051462224

J. 0.9925461516

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교과서 질문

CONCEPT PREVIEW Match the measure of bearing in Column I with the appropriate graph in Column II.

I. S 70° W

II. 1. A. B. C. 2. S 70° W 3. 4. D. E. F. 5. 6. 7. G. H. 8. 9. 10. I. J.

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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

697

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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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