91Ó°ÊÓ

In mathematics, a quadratic equation is any equation that can be rearranged in standard form as \( ax^2 + bx + c = 0 \), where \( x \) represents an unknown variable, and \( a, b, \) and \( c \) are constants with \( a eq 0 \). The term \( ax^2 \) is referred to as the quadratic term, \( bx \) the linear term, and \( c \) the constant term. If \( a = 0 \), the equation becomes linear, not quadratic.

The solutions to a quadratic equation can be determined in various ways:

• Factoring: If the quadratic expression on the left side can be factored into the product of two binomials.
• Completing the Square: Making the quadratic expression into a perfect square trinomial.
• Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), useful for any quadratic equation.

In the context of the problem, the equation \(2 \sin^2 \theta - \sin \theta - 1 = 0\) can be seen as a quadratic equation by treating \( \sin \theta \) as \( x \). This substitution allows us to apply quadratic methods to solve it.

The sine function is a fundamental trigonometric function, which is essential for studying the properties of triangles and the relationship between the angles and lengths of those triangles. Specifically, the sine function relates a given angle \( \theta \) in a right-angled triangle to the ratio of the length of the side opposite to \( \theta \) to the hypotenuse.

Sine function has a range between -1 and 1, which plays a crucial role when solving trigonometric equations as any result beyond this does not fit a real angle value.

• Common values: Examples include \( \sin(0) = 0 \), \( \sin(\frac{\pi}{2}) = 1 \), and \( \sin(\pi) = 0 \).
• Periodicity: The sine function is periodic with a period of \(2\pi\), meaning \( \sin(\theta) = \sin(\theta + 2k\pi) \) for any integer \(k\).

Understanding these properties helps us solve the equation \( \sin \theta = x \), where \( x \) can only be any value between -1 and 1, such as in our problem.

Finding the general solution of a trigonometric equation involves determining all possible values of the variable that satisfy the equation. Given the periodic properties of trigonometric functions, general solutions take the form of a basic solution plus an integer multiple of the function's period.

For example, if \( \sin \theta = -\frac{1}{2} \), we know this angle and related angles will repeat every complete rotation (or \(2\pi\)).

• For \( \sin \theta = -\frac{1}{2} \), typical solutions include \(\theta = -\frac{\pi}{6} + 2k\pi\) and \(\theta = \frac{7\pi}{6} + 2k\pi\).
• Similarly, for \( \sin \theta = 1 \), we have \(\theta = \frac{\pi}{2} + 2k\pi\).

By adding multiples of \(2\pi k\), where \(k\) is an integer, we account for the infinite repeats of sine's behavior over its cycle, ensuring we do not miss any potential solutions.

Factoring is a core method for solving quadratic equations. It involves breaking down the quadratic expression into a product of simpler expressions set to zero, thus finding its roots.

In the equation \(2x^2 - x - 1=0\):

• Seek two numbers that multiply to \( -2 \) (the product of the coefficient of \( x^2 \) and the constant term) and add to \(-1\) (the coefficient of \( x \)). These numbers are \(-2\) and \(1\).
• Rewrite \(-x \) as \(-2x + x\).
• Group terms and factor by grouping: \( (2x^2 - 2x) + (x - 1) \), which becomes \(2x(x-1) + 1(x-1)\).
• Factor out the common terms: \((2x + 1)(x - 1) = 0\).

This results in the simpler equations \(2x + 1 = 0\) or \(x - 1 = 0\), allowing us to directly solve for the variable \( x \), completing the problem-solving process.