91Ó°ÊÓ

A quadratic equation is a second-order polynomial equation in a single variable. Generally, it has the form \(ax^2 + bx + c = 0\).
Here, \(a\), \(b\), and \(c\) are constants, where \(a eq 0\).
In our problem, the trigonometric equation is transformed into a quadratic form: \(x^2 - 4x + 2 = 0\), where \(x\) represents \(\text{cot}(\theta)\). Understanding quadratic equations is crucial because they appear frequently in mathematics.
They allow us to find unknown values by solving for the variable.
Here are the primary steps to solve a quadratic equation:

• Identify coefficients: Look for the values of \(a\), \(b\), and \(c\) in the equation.
• Calculate the discriminant: It's given by \(b^2 - 4ac\). This helps determine the number and type of solutions.
• Use the quadratic formula: Plug the values of \(a\), \(b\), and \(c\) into \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].
  This will give you the roots of the equation.

In our exercise, after calculating the discriminant and applying the formula, we find the roots: \(x = 2 + \sqrt{2}\) or \(x = 2 - \sqrt{2}\). These values are critical for solving the trigonometric equation, as they represent \(\text{cot}(\theta)\).

Inverse trigonometric functions allow us to find angles when given a trigonometric value. For example, if we know \(\text{cot}(\theta)\), we can find \(\theta\) using the inverse cotangent function, denoted as \( \cot^{-1}(x) \).
These functions are essential in trigonometry because they reverse the roles of output and input.
Here’s a quick look at how it's done in our exercise:

• Identify the values: From the quadratic equation, solve for \(x\) to get \(2 + \sqrt{2}\) and \(2 - \sqrt{2}\).
• Use the inverse function: Apply \( \cot^{-1}(x) \) to find the angles corresponding to these values.
  So, we compute \(\theta = \cot^{-1}(2 + \sqrt{2})\) and \( \theta = \cot^{-1}(2 - \sqrt{2})\).
• Round the results: Since we need the values in degrees, we round them to the nearest tenth of a degree.

Inverse trigonometric functions are crucial for converting between angle measures and trigonometric values.In our case, they help us determine specific angles \(\theta\) within the interval \([0^{\circ}, 360^{\circ}]\), which satisfies the original trigonometric equation.

The cotangent function, denoted as \( \text{cot}(\theta) \), is one of the basic trigonometric functions. It's the reciprocal of the tangent function: \( \text{cot}(\theta) = \frac{1}{\text{tan}(\theta)} \).
This means cotangent gives the ratio of the adjacent side to the opposite side in a right triangle.
Here are a few key points to understand the cotangent function better:

• Range and Domain: The cotangent function is defined for all angles, except where \(\text{tan}(\theta) = 0\) (i.e., multiples of \(\frac{\pi}{2}\) radians, or \(90^{\circ}\)).
• Graph behavior: Since \(\text{cot}(\theta)\) is periodic, it repeats its values in cycles of \(180^{\circ}\).
• Properties and Identities: It helps to know identities such as \(\text{cot}(\theta) = \text{cos}(\theta) / \text{sin}(\theta)\).
  This can simplify solving trigonometric equations.

In our problem, understanding the cotangent function allows us to substitute appropriately and later use its inverse to find all possible angle values.
We utilized the fact that \( \text{cot}(\theta) = 2 + \sqrt{2} \) or \( 2 - \sqrt{2} \) and then found the angles \(\theta\) using the inverse cotangent function.
This entire process hinges on a good grasp of the cotangent and its properties.