Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problem 2.3.15

Chapter 3, Problem 2.3.15

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'

Verified step by step guidance

First, convert the angle from degrees and minutes to a decimal degree format. Since 1 minute is \( \frac{1}{60} \) of a degree, convert 48' to degrees by calculating \( 48 \times \frac{1}{60} \). Then add this to 183° to get the total angle in decimal degrees.

Express the cotangent function in terms of sine and cosine to understand its relationship: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). This helps in understanding the behavior of the function if needed.

Use a calculator to find the sine and cosine of the decimal degree angle. Make sure your calculator is set to degree mode, not radians.

Calculate the cotangent by dividing the cosine value by the sine value: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Round the result to six decimal places as required.

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Key Concepts

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

Angle Conversion from Degrees and Minutes to Decimal Degrees

Angles given in degrees and minutes must be converted to decimal degrees before calculation. Since 1 minute equals 1/60 of a degree, convert minutes by dividing by 60 and add this to the degrees. For example, 183° 48' equals 183 + 48/60 = 183.8°.

Cotangent Function and Its Relationship to Sine and Cosine

The cotangent of an angle is the reciprocal of the tangent function, or equivalently, cot(θ) = cos(θ)/sin(θ). Understanding this relationship helps simplify expressions and interpret results, especially when using a calculator that may not have a direct cotangent function.

Using a Calculator for Trigonometric Approximations

Calculators typically require angles in decimal degrees or radians and may not have a cotangent button. After converting the angle, use the reciprocal of tangent or cosine over sine to find cotangent. Round the final answer to the specified decimal places, here six, for precision.

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Textbook Question

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

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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

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.

tan 421° 30'

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Textbook Question

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. (-4, 0)

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Textbook Question

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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