Problem 46 - Algebra 2 and Trigonometry

Question

In \(39-50,\) find the smallest positive value of \(\theta\) to the nearest degree. $$ \tan \theta=1.8808 $$

Step-by-Step Solution

Answer

The smallest positive value of \( \theta \) is approximately 62 degrees.

Step 1: Understanding the Problem

We are given the equation \( \tan \theta = 1.8808 \) and need to find the smallest positive value of \( \theta \). The angle \( \theta \) should be rounded to the nearest degree.

Step 2: Solve for \( \theta \)

To solve for \( \theta \), we use the inverse tangent function, denoted as \( \theta = \tan^{-1}(1.8808) \). This function will give us the angle whose tangent is 1.8808.

Step 3: Calculate Using a Calculator

Using a calculator, we find \( \theta = \tan^{-1}(1.8808) \approx 61.7 \) degrees. Calculators typically provide angles in degrees or radians, ensure your calculator is set to degrees.

Step 4: Round \( \theta \) to the Nearest Degree

Finally, we round \( 61.7 \) degrees to the nearest whole number, which results in \( 62 \) degrees.

Key Concepts

Inverse Tangent Function

The inverse tangent function, also known as the arctangent function, is a crucial concept in trigonometry. It allows us to find the angle whose tangent value is given. The notation for the inverse tangent function is typically written as \( \tan^{-1}(x) \) or \( \text{arctan}(x) \). This can sometimes be a bit confusing because it seems like we might be dividing by the tangent function, but instead, we're actually applying the process of finding the angle.

• **What it does**: The function outputs the angle \( \theta \) in such a way that \( \tan(\theta) = x \).
• **Range of Output**: Usually, the range of the inverse tangent function is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians, or in degrees, from \(-90^\circ\) to \(90^\circ\).
• **Usage**: Whenever you know the tangent of an angle but need to determine the angle itself, this function is your tool of choice.

When you have a calculator, finding \( \theta \) for a given tangent involves entering the value into the inverse tangent function. This process immediately gives you the desired angle.To find the angle corresponding to a specific tangent value, ensure that your calculator is set to the correct mode (degrees or radians). For example, to find an angle in degrees, inputting \( \tan^{-1}(1.8808) \) in a calculator yields approximately 61.7 degrees.

Tangent of an Angle

The tangent of an angle in a right triangle is a fundamental trigonometric ratio. It's the ratio of the length of the opposite side to the length of the adjacent side. This trigonometric function helps to relate the angle \( \theta \) to the dimensions of the triangle. Understanding this ratio is essential for different applications, such as solving geometric problems and calculating angles.

• **Definition**: In any right triangle, the tangent of an angle \( \theta \) is given by the formula: \( \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \).
• **Periodic Nature**: The tangent function is periodic with a period of \(180^\circ\) or \(\pi\) radians, meaning it repeats its values every \(180^\circ\).
• **Properties**: Tangent values can be any real number, but for angles greater than \(90^\circ\), the interpretations and calculations need careful handling.

The tangent function provides a handy way to solve for angles when side lengths are known because trigonometric relationships are used to calculate unknown angles. This foundational knowledge is vital for using the inverse tangent function and working with other trigonometric identities.For example, if you know that \( \tan(\theta) = 1.8808 \), this means the opposite side is 1.8808 times the length of the adjacent side. This ratio is crucial for situational geometry and calculus applications.

Rounding to Nearest Degree

Rounding is a simple yet important step in many calculations. It helps simplify results and make them easier to interpret, especially when working with measured or calculated angles. Rounding to the nearest degree means adjusting a decimal angle value to the closest whole number. When you have a decimal like 61.7 degrees, here's how to round it correctly:

• **Look at the tenths place digit**: If it's 5 or more, round up. If it’s less than 5, round down.
• **Example**: For 61.7, the tenths place is 7, which is greater than 5. So, we round up to 62 degrees.
• **Purpose of rounding**: Provides clarity and simplicity in reporting measurements or when applying specific angles in practical scenarios like construction or navigation.

Rounding is vital in contexts where shared understanding and straightforward communication about angles are necessary like in education or engineering. It ensures that everyone interprets results consistently without the potential misunderstanding that could arise from more complex decimal numbers.