Problem 23 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \tan 88^{\circ} 30^{\prime} $$

Step-by-Step Solution

Answer

\(\tan 88^{\circ} 30^{\prime} \approx 41.6336\).

Step 1: Understand the Problem

We need to find the tangent of the angle 88 degrees and 30 minutes, which is written as \(88^{\circ} 30^{\prime}\). We will convert this angle into a decimal degree format and then use a calculator to find the tangent value.

Step 2: Convert Minutes to Decimal Degrees

Each degree is divided into 60 minutes. To convert minutes into decimal degrees, we divide the minutes by 60. Hence, we have \(30\) minutes to convert: \(30 \div 60 = 0.5\) degrees.

Step 3: Calculate Total Degrees

Add the degrees from the first part of the angle to the decimal degrees we obtained from converting minutes: \(88 + 0.5 = 88.5\) degrees.

Step 4: Evaluate the Tangent Function

Use a calculator to find \(\tan(88.5^{\circ})\). Ensure the calculator is in degree mode, and then compute the tangent of 88.5 degrees.

Step 5: Round the Answer

The answer from the calculator will likely be a long decimal. We need to round the result to four decimal places for the final answer.

Key Concepts

Angle Conversion

When working with trigonometric functions, one key step is converting angles to a format that can be easily calculated. Angles may appear in different formats, such as degrees and minutes.
To convert an angle from degrees and minutes into decimal degrees, you need to convert the minutes first. Each degree is divided into 60 equal parts called minutes, symbolized by the apostrophe ('). For instance, 1 degree equals 60 minutes.
To convert minutes into decimal form, divide the number of minutes by 60. If you encounter an angle like 88° 30', divide 30 by 60 to get 0.5.

• Calculation:
  • Convert 30 minutes: 30 ÷ 60 = 0.5 degrees
• Add this result to the original degree measure: 88° + 0.5° = 88.5°

Once you have the angle in decimal degrees, you can perform further calculations, like finding its trigonometric function values.

Tangent Calculation

The tangent function, a primary trigonometric function, is essential in the study of angles and lengths in right triangles. It is defined as the ratio of the opposite side to the adjacent side of a right triangle. In notation, it is represented by \(\tan\).

To find the tangent of an angle, like 88.5 degrees, you will usually need a scientific calculator. First, ensure the calculator is set to degree mode because angles can also be measured in radians.

• Set your calculator to degree mode.
  • This ensures that any angle input is understood as degrees.
• Enter the degree measure: 88.5.
• Calculate \(\tan(88.5^{\circ})\).

The calculator will provide a numeric value for the tangent of the given angle. This process highlights the utility of the tangent function in trigonometry.

Rounding Decimals

After calculating trigonometric functions, like the tangent, you often encounter lengthy decimal values. These need to be rounded, particularly when instructed to provide answers to a certain decimal place accuracy.

Rounding decimals is adjusting the number to a specified degree of precision, often linked to the number of decimal places. For this task, you’ll typically round to four decimal places.

• Identify the digit in the fourth decimal place.
  • This is the limit of precision requested.
• Look at the digit right next to it (the fifth decimal place).
  • If this digit is 5 or greater, round up the fourth decimal place digit by one.
  • If it is less than 5, leave the fourth decimal place digit unchanged.

By rounding the decimals from the tangent calculation, you tidy up the results, making them neater and easier to interpret in further uses.