Problem 22 - Precalculus: Functions and Graphs

Question

Approximate to three decimal places. (a) cot \(9^{\circ} 10^{\prime}\) b) \(\tan 0.75\)

Step-by-Step Solution

Answer

(a) 6.245; (b) 0.013

Step 1: Convert Degrees and Minutes to Decimal Degrees

For part (a), we first convert the angle given in degrees and minutes to decimal degrees. There are 60 minutes in one degree, so:\[ 9^{\circ} 10' = 9 + \frac{10}{60} = 9.1667^{\circ} \]

Step 2: Use a Calculator for Cotangent

Now, find the cotangent of the angle we just converted. Cotangent is the reciprocal of tangent, so:\[ \cot 9.1667^{\circ} = \frac{1}{\tan 9.1667^{\circ}} \]Use a calculator to find \( \tan 9.1667^{\circ} \) and then take its reciprocal to get \( \cot 9.1667^{\circ} \), approximating to three decimal places.

Step 3: Calculate Tangent for Part (b)

For part (b), we directly compute the tangent of 0.75 degrees using a calculator:\[ \tan 0.75^{\circ} \]Make sure your calculator is set to degrees and approximate the value to three decimal places.

Key Concepts

Cotangent

Cotangent is a fundamental trigonometric function. It is very closely related to the tangent function, being its reciprocal. This means that the cotangent of an angle is calculated as the inverse of the tangent of the same angle. The formula is:

• \( \cot \theta = \frac{1}{\tan \theta} \)

This relationship makes cotangent particularly useful when solving trigonometric problems that involve right triangles or periodic functions. To find the cotangent of any angle, you can first calculate the tangent and then simply take its reciprocal.
In a practical scenario, especially when working with non-unit angles like \(9.1667^{\circ}\), using a calculator becomes essential. You would compute the tangent first and then divide one by this result to get the cotangent approximated to the desired decimal places.

Tangent

The tangent function is one of the primary trigonometric functions, commonly used in geometry and calculus. Its main function is to relate an angle in a right triangle to the ratio of the length of the opposite side to the adjacent side:

• \( \tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} \)

In more advanced applications, the tangent is pivotal in defining the behavior of periodic functions. To compute \( \tan \theta \) for any given angle \( \theta \), a calculator set to degrees is usually employed, ensuring accurate results. For example, finding \( \tan 0.75^{\circ} \) means you'd directly use a calculator to obtain a result approximated to three decimal places. This simplicity makes tangent practical for a wide range of scientific applications.

Angle Conversion

Angle conversion is a common necessity when dealing with trigonometric problems, particularly when angles are given in degrees and minutes. Since most trigonometric functions require angles in decimal degrees, you must know how to convert from degrees and minutes to a decimal format. The conversion process involves understanding that there are 60 minutes in one degree. To convert:

• Add the whole degrees as they are.
• Convert the minutes by dividing them by 60.
• Sum these to obtain the angle in decimal degrees.

For instance, an angle of \(9^{\circ} 10'\) is converted to decimal degrees using the formula \( 9^{\circ} + \frac{10}{60} = 9.1667^{\circ} \). This precision is crucial, especially when high accuracy is required in results.

Decimal Degrees

Decimal degrees are a simple and standardized way of representing angles, making calculations easier when compared to using degrees and minutes. The use of decimal degrees allows for straightforward arithmetic operations, giving precision to trigonometric calculations expected in many scientific and mathematical contexts. To convert traditional angles given in minutes to decimal degrees:

• Understand that one degree is divided into 60 minutes.
• To convert, divide the minutes by 60 to turn them into a fraction of a degree.
• Add this fraction to the whole degrees to get the decimal degree.

Decimal degrees are particularly helpful when entering data into calculators or computer software, which typically do not accept the complex degree-minute-second format. They help ensure results are accurate and formatted correctly for subsequent mathematical operations.