91Ó°ÊÓ

Inverse trigonometric functions help us find the angles when their trigonometric ratios are known. For example, if we know the tangent of an angle, we can use the inverse tangent function, often written as \( \tan^{-1} \), to find the angle itself.
These functions are crucial when solving equations involving trigonometric expressions, especially when we're asked to find angles in specific ranges.
When working with the tangent, remember that the inverse tangent function yields angles typically in the interval \(( -\frac{\pi}{2}, \frac{\pi}{2} )\). This range ensures that the tangent is one-to-one, allowing us to get a unique angle for each value of \( y = \tan x \).
In this exercise, after identifying possible values of \( \tan x \), we used \( \tan^{-1} \) to determine the angles \( x \) that solved our original equation within the given interval.

A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c = 0 \). Solving quadratic equations is essential in mathematics, as they appear in various contexts, including physics, engineering, and economics.
To solve these equations, we can use different methods like factorization, completing the square, or the quadratic formula. In this exercise, we used the quadratic formula to solve the equation \( y^2 - 3y + 1 = 0 \), where \( y \) represented \( \tan^2 x \).
The quadratic formula is given by \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, if you substitute \( a = 1, b = -3, \) and \( c = 1 \), you derive the solutions for \( y \). Understanding the roots of this equation allowed us to determine potential values for \( \tan x \).

The tangent function, expressed as \( \tan x \), is a basic trigonometric function that relates an angle to the ratio of the length of the opposite side to the adjacent side in a right triangle.
It's periodic and has a period of \( \pi \), meaning it repeats its values every \( \pi \) radians. In addition, it has vertical asymptotes every \( \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
In this exercise, we manipulated the equation involving \( \tan^4 x \) and \( \tan^2 x \), substituting \( y = \tan^2 x \) to simplify and solve it as a quadratic equation. This allowed us to find possible values of \( \tan x \) that could satisfy the original equation.
The solutions we found for \( \tan x \) were used to identify the angle values with the help of the inverse tangent function, thus solving for \( x \).

Interval notation offers a concise way to represent a range of values between two endpoints. It is particularly useful for expressing solutions to inequalities and functions.
In mathematics, interval notation uses brackets and parentheses to represent closed and open endpoints, respectively. For example, \([a, b]\) denotes an interval including the endpoints \(a\) and \(b\), whereas \((a, b)\) excludes them.
In our exercise, the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\) was given, indicating that the solutions had to fall between these two values but not include them. This ensured that any angles we found were within the valid domain of the inverse tangent function.
Verifying that all obtained angle solutions were within the specified interval confirmed their validity as solutions to the original trigonometric equation.