Problem 41 - Algebra 2 and Trigonometry

Question

In \(39-50,\) find the smallest positive value of \(\theta\) to the nearest degree. $$ \tan \theta=0.2126 $$

Step-by-Step Solution

Answer

\(\theta \approx 12^\circ\)

Step 1: Understanding the Problem

We need to find the smallest positive angle, \( \theta \), that satisfies the equation \( \tan \theta = 0.2126 \). The tangent function is periodic, so there are multiple solutions for \( \theta \). Our focus is to find the angle in degrees between \(0^\circ\) and \(360^\circ\).

Step 2: Use the Inverse Tangent Function

To find \( \theta \), we use the inverse tangent function (also known as arctangent). We calculate \( \theta = \tan^{-1}(0.2126) \), which will give us an angle in radians, typically in the first quadrant for a positive value of the tangent.

Step 3: Convert Radians to Degrees

After computing \( \theta = \tan^{-1}(0.2126) \), we convert the result from radians to degrees. We use the conversion factor where \(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\).

Step 4: Perform the Calculation

Compute \( \theta = \tan^{-1}(0.2126) \), which gives a value in radians. For example, if \( \theta \approx 0.2093 \text{ radians} \). Convert this to degrees: \(0.2093 \times \frac{180}{\pi} \approx 12.00 \).

Step 5: Round to the Nearest Degree

After converting to degrees, round the angle to the nearest whole number. From the calculation, \( \theta \approx 12^\circ \). Therefore, the smallest positive angle satisfying \( \tan \theta = 0.2126 \) is \(12^\circ\).

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \theta \), is one of the primary trigonometric functions. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. Understanding the tangent function is crucial because it helps relate an angle to lengths in a triangle.

• The tangent function has a range of all real numbers, which means it can output any value from negative infinity to positive infinity.
• It is undefined for angles where the cosine is zero, such as at \(90^\circ\) and \(270^\circ\) in the unit circle.
• The function exhibits a repeating pattern known as periodicity, which means its values repeat at regular intervals of \(180^\circ\) (or \(\pi\) radians).

To solve an equation like \(\tan \theta = 0.2126\), we use the inverse tangent function, denoted as \( \tan^{-1} \) or \( \arctan \), which helps find the angle \( \theta \) that corresponds to a given tangent value.

Angle Conversion

Angle conversion is essential when working with trigonometric functions, as angles can be measured in degrees or radians. Mathematicians and scientists often prefer radians, but degrees are more intuitive for everyday use.

• Degrees and radians are two different units for measuring angles. There are \(360^\circ\) in a full circle and \(2\pi\) radians in the same circle.
• To convert from radians to degrees, use the formula \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \).
• Conversely, to convert degrees to radians, use \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).

When finding \(\theta\) from the tangent value, we often get the result in radians. Converting this result to degrees ensures it is more understandable, especially when identifying angles within a typical range from \(0^\circ\) to \(360^\circ\).

Periodicity of Trigonometric Functions

The periodicity of trigonometric functions is a crucial concept when solving trigonometric equations, as it describes how the function's values repeat over specific intervals.

• For the tangent function, the key is its periodicity of \(180^\circ\) (or \(\pi\) radians), which means \( \tan(\theta) = \tan(\theta + n \cdot 180^\circ) \) for any integer \( n \).
• This knowledge helps determine multiple solutions for an equation like \( \tan \theta = 0.2126 \).
• We calculate the smallest positive angle first and be aware that adding any multiple of \(180^\circ\) will yield another valid angle of the same tangent value.

By understanding periodicity, we can confidently locate all possible solutions within a given range and verify the smallest positive solution, such as finding \( \theta \approx 12^\circ \) within the interval \(0^\circ\) to \(360^\circ\).