91Ó°ÊÓ

Quadratic equations can sometimes appear in trigonometric problems like the one discussed in our exercise. In this particular example, our equation was transformed into a quadratic form relating to \( \cos 2x \). A quadratic equation typically takes the form \( ax^2 + bx + c = 0 \). These equations can have two solutions, which are found using various methods, such as factoring, completing the square, or the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

In this problem, the trigonometric identity conversions eventually led to a quadratic equation in terms of \( \cos 2x \). Solving it involved manipulating the terms into the standard quadratic form to find solutions using the quadratic formula. The resultant solutions give potential values for \( \cos 2x \), which are crucial for further steps in finding the solutions to the trigonometric equation.

Inverse trigonometric functions allow us to determine angles when given the values of trigonometric functions like sine, cosine, or tangent. Given a trigonometric ratio, inverse functions help to find the corresponding angle.

For example, if we know that \( \cos 2x = a \), we can find \( 2x \) by calculating \( \cos^{-1}(a) \). This process is an essential step in solving trigonometric equations. It's important to remember the ranges of inverse functions, which limit their possible outputs to ensure they are functions.

• \( \sin^{-1}(x) \) produces values in \([-\pi/2, \pi/2]\)
• \( \cos^{-1}(x) \) gives results in \([0, \pi]\)
• \( \tan^{-1}(x) \) has an output in \((-\pi/2, \pi/2)\)

In our exercise, using \( \cos^{-1} \), we found the angle corresponding to each solution from the quadratic equation, further helping us to solve for \( x \).

Solving trigonometric equations involves finding all angles \( x \) that satisfy the equation. In this context, after obtaining \( \cos 2x \) values from our quadratic solution, the challenge shifted to finding the actual \( x \). This typically involves:

• Identifying possible solutions using inverse trigonometric functions.
• Dividing the results by any multipliers in the argument, like the 2 in \( 2x \).
• Considering the periodic nature of trigonometric functions by adding multiples of their periods to find all possible solutions.

By following these procedures, we ensure that the correct set of solutions for \( x \) are obtained. Checking these solutions in the original equation helps validate their correctness and completeness.