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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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

Find a value of θ, in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. sec θ = 1.2637891

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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°