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Ch. 2 - Acute Angles and Right Triangles
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 2 - Acute Angles and Right TrianglesProblem 2.3.22
Chapter 3, Problem 2.3.22

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.
1/ sec 14.8°

Verified step by step guidance
1
Recall the definition of the secant function: \(\sec \theta = \frac{1}{\cos \theta}\). This means that \(\frac{1}{\sec \theta} = \cos \theta\).
Rewrite the given expression \(\frac{1}{\sec 14.8^\circ}\) as \(\cos 14.8^\circ\) using the identity from step 1.
Use a calculator to find the value of \(\cos 14.8^\circ\). Make sure your calculator is set to degree mode since the angle is given in degrees.
Calculate the cosine value and round the result to six decimal places as requested.
Write down the final answer with six decimal places, ensuring the approximation is clear and precise.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Understanding the Secant Function

The secant function, sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). Knowing this relationship allows you to rewrite expressions involving secant in terms of cosine, which is often easier to evaluate using a calculator.
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Graphs of Secant and Cosecant Functions

Simplifying Trigonometric Expressions

Simplifying expressions before calculation helps reduce errors and makes the evaluation process straightforward. For example, rewriting 1/sec(θ) as cos(θ) simplifies the expression and avoids dealing with complex reciprocal values directly.
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Using a Calculator for Trigonometric Values

Calculators typically provide trigonometric functions like sine, cosine, and tangent. To find values like cos(14.8°), ensure the calculator is set to degree mode, input the angle correctly, and round the result to the required decimal places for precision.
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Related Practice
Textbook Question

Solve each problem. See Examples 1 and 2. Distance Traveled by a Ship A ship travels 55 km on a bearing of 27° and then travels on a bearing of 117° for 140 km. Find the distance from the starting point to the ending point.

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Textbook Question

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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Textbook Question

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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Textbook Question

Find exact values of the six trigonometric functions for each angle A.

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Textbook Question

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

I. 8. 270°


II.

1. A. B. C. 2. 3. 4. D. E. F. 5. 6. 7. G. H. 9. 10. I. J.

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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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