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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Ch. 2 - Acute Angles and Right Triangles
Lial12th EditionTrigonometryISBN: 9780136552161
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TOC.tableOfContents
- 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
3장, 문제 2.3.34
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
검증된 단계별 안내1
Recall the definition of the cosecant function: \(\csc \theta = \frac{1}{\sin \theta}\). This means that \(\sin \theta = \frac{1}{\csc \theta}\).
Substitute the given value of \(\csc \theta\) into the equation: \(\sin \theta = \frac{1}{1.3861147}\).
Calculate the value of \(\sin \theta\) using the reciprocal of \(1.3861147\) (do not compute the final decimal here, just set up the expression).
Use the inverse sine function to find \(\theta\): \(\theta = \sin^{-1}(\sin \theta)\), where \(\sin \theta\) is the value found in the previous step.
Since the problem restricts \(\theta\) to the interval \([0^\circ, 90^\circ)\), select the principal value of \(\theta\) from the inverse sine calculation and express it in decimal degrees to six decimal places.
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.
Cosecant Function (csc θ)
The cosecant function is the reciprocal of the sine function, defined as csc θ = 1/sin θ. It is undefined when sin θ = 0 and is used to find angles when the ratio of the hypotenuse to the opposite side is known.
추천 영상:
6:22
Graphs of Secant and Cosecant Functions
Inverse Trigonometric Functions
Inverse trigonometric functions allow us to find an angle when given a trigonometric ratio. For csc θ, we first find sin θ by taking the reciprocal, then use the inverse sine (arcsin) to determine θ within the specified interval.
추천 영상:
4:28Introduction to Inverse Trig Functions
Angle Measurement and Interval Restrictions
Angles are measured in degrees or radians, and specifying an interval like [0°, 90°) restricts the solution to the first quadrant. This ensures the angle found is within the domain where sine and cosecant are positive.
추천 영상:
5:31Reference Angles on the Unit Circle
관련 실천
교과서 질문
Use a calculator to evaluate each expression. 2 sin 25°13' cos 25°13' - sin 50°26'
675
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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'
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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
(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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