Problem 57 - Algebra 2 and Trigonometry

Question

In \(51-58,\) find the smallest positive value of \(\theta\) to the nearest minute. $$ \tan \theta=7.3478 $$

Step-by-Step Solution

Answer

The smallest positive value of \(\theta\) is approximately \(82\) degrees and \(15\) minutes.

Step 1: Understand the Problem

We need to find an angle \(\theta\) where the tangent of the angle equals 7.3478. The objective is to find the smallest positive angle in degrees and convert that angle to minutes.

Step 2: Calculate the Angle Using the Inverse Tangent

Use the inverse tangent function to calculate \(\theta\). In a calculator, input \(\theta = \tan^{-1}(7.3478)\). This yields \(\theta \approx 82.2437\) degrees.

Step 3: Convert the Decimal Degrees to Degrees and Minutes

Separate the integer part and the decimal part of the degrees. We have \(82\) degrees and \(0.2437\) of a degree remaining. To convert \(0.2437\) degrees to minutes, multiply by 60: \(0.2437 \times 60 \approx 14.622\) minutes.

Step 4: Round the Minutes to the Nearest Whole Number

Since \(14.622\) minutes is closest to \(15\) minutes, we round up to \(15\) minutes.

Key Concepts

Tangent Function

The tangent function is one of the six fundamental trigonometric functions in mathematics. It relates an angle of a right triangle to the ratio of the length of its opposite side to the length of its adjacent side. Essentially, for a given angle \( \theta \), the tangent is represented as:\[\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}.\]In our particular problem, we are given \( \tan \theta = 7.3478 \). This suggests that for the angle \( \theta \), the opposite side is approximately 7.3478 times longer than the adjacent side. Our goal here is to find \( \theta \), which is achieved through the inverse tangent function.

Angle Calculation

To find angle \( \theta \) given \( \tan \theta = 7.3478 \), we use the inverse tangent function, often notated as \( \tan^{-1} \) or "arctan" in calculators.Steps:

• Enter the value \( 7.3478 \) into your calculator.
• Apply the inverse tangent function, typically labeled as "tan⁻¹" or "arctan".

This process yields an angle in degrees. For this particular example, this computation results in an angle of approximately 82.2437 degrees. Always remember that for this function, the output is the angle \( \theta \) whose tangent equals the input number.

Degree to Minute Conversion

When dealing with angles, degrees are often further divided into smaller units called minutes. One degree equals 60 minutes. After calculating an angle in decimal degrees, it's common to convert the decimal portion into minutes for precision.Steps:

• Observe the decimal part of the angle: In our example, we've calculated \( 82.2437 \) degrees.
• Separate this into \( 82 \) degrees and \( 0.2437 \) degrees.
• Convert \( 0.2437 \) degrees to minutes by multiplying by 60, as 1 degree = 60 minutes.

For our case, \( 0.2437 \times 60 \approx 14.622 \) minutes. Hence, \( \theta \) is approximately \( 82 \) degrees and \( 14.622 \) minutes before rounding.

Rounding

Rounding involves adjusting numbers to a specified degree of precision. It's particularly useful when converting between units, like from decimal degrees to minutes, to make values more comprehensible without unnecessary detail.In our problem, after converting \( 0.2437 \) degrees to \( 14.622 \) minutes, we need to round \( 14.622 \) to the nearest whole number for ease of communication.Steps:

• If the decimal is 0.5 or greater, round up. Otherwise, round down.

Here, since \( 14.622 \) is closer to 15, we round up to 15 minutes. Thus, the smallest positive angle \( \theta \) we are looking for is \( 82 \) degrees and \( 15 \) minutes.