Problem 22 - Algebra 2 and Trigonometry

Question

In \(13-24,\) find, to the nearest ten-thousandth, the radian measure \(\theta\) of a first-quadrant angle with the given function value. $$ \cot \theta=0.3276 $$

Step-by-Step Solution

Answer

The radian measure of \(\theta\) is approximately \(1.2505\).

Step 1: Understand the Relationship

In trigonometry, the cotangent of an angle \(\theta\) is the reciprocal of the tangent. So, \(\cot \theta = \frac{1}{\tan \theta}\). Given that \(\cot \theta = 0.3276\), you can find \(\tan \theta\) as \(\tan \theta = \frac{1}{0.3276}\).

Step 2: Calculate the Tangent Value

Use the reciprocal relationship to find \(\tan \theta\): \[\tan \theta = \frac{1}{0.3276} \approx 3.0533\]

Step 3: Use Inverse Tangent Function

To find the angle \(\theta\), use the inverse tangent function (also known as arctan or \(\tan^{-1}\)) on a calculator: \[\theta = \tan^{-1}(3.0533)\].

Step 4: Convert the Angle to Radians

When calculating \(\theta = \tan^{-1}(3.0533)\), you get approximately \(\theta \approx 71.63^\circ\). Convert this degree measure to radians using the formula \(\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}\).

Step 5: Perform the Calculation

Convert \(71.63^\circ\) to radians: \(\theta_{\text{radians}} = 71.63 \times \frac{\pi}{180} \approx 1.2505\) radians.

Step 6: Round to the Nearest Ten-Thousandth

Round the radian measure \(1.2505\) to the nearest ten-thousandth. The rounded radian measure is \(1.2505\).

Key Concepts

Cotangent

The cotangent of an angle, often abbreviated as "cot," is a fundamental concept in trigonometry. It is defined as the reciprocal of the tangent function. This means that \[\cot \theta = \frac{1}{\tan \theta}\]where \( \theta \) is the angle in question. When you have a cotangent value, like 0.3276 in this problem, you can easily find the tangent by taking the reciprocal: \[\tan \theta = \frac{1}{0.3276} \approx 3.0533\]The cotangent function is especially useful in cases where you need to find the corresponding tangent value quickly. Knowing the relationship between cotangent and tangent helps simplify many trigonometric calculations.

Inverse Trigonometric Functions

Inverse trigonometric functions allow you to find angles when you know the value of a trigonometric function. For the tangent function, the inverse is commonly denoted as \(\tan^{-1}\) or \(\arctan\). In this problem, once the tangent of the angle is found as 3.0533, the next step is to use this inverse function: \[\theta = \tan^{-1}(3.0533)\]This will provide the angle \( \theta \) in degrees. Inverse trigonometric functions are vital for converting between angle measures and trigonometric values, making them an essential tool in trigonometry.

Radians

To fully grasp trigonometry, it's crucial to understand the concept of radians. Degrees are a common unit for measuring angles, but in mathematics, particularly in calculus, radians are often used for accuracy and simplicity. One complete revolution in a circle is \(2\pi\) radians, which is equivalent to 360 degrees. To convert an angle from degrees to radians, you can use the formula: \[\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}\]In this exercise, after finding the angle in degrees as approximately 71.63, it is converted to radians as follows: \[\theta_{\text{radians}} = 71.63 \times \frac{\pi}{180} \approx 1.2505\]Understanding radians is essential for working with trigonometric functions and for performing various calculations in advanced mathematics.