91Ó°ÊÓ

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). The solutions to this equation are called the "roots". They are the values of \(x\) that make the equation true, meaning they make the expression equal to zero. Finding these roots can tell us a lot about an equation's graph and its relationships.

For any quadratic equation, the roots can be calculated using the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here, the expression under the square root \((b^2 - 4ac)\) is called the discriminant. The sign and value of the discriminant help us understand the nature of the roots:

• A positive discriminant indicates two real and different roots.
• A discriminant of zero signals one real, repeated root.
• A negative discriminant means no real roots, rather two complex roots.

For our problem, understanding that 1 lies between the roots gives us a specific inequality to solve, which can lead us to explore the relationship between the equation's coefficients and the substitution used.

Trigonometric functions, like sine and cosine, offer exciting ways to transform equations, especially when dealing with cyclical patterns or angles. In the given exercise, we express \(\cos^2\theta\) using the identity \(\cos^2\theta = 1 - \sin^2\theta\). This transformation allows us to replace \(\cos^2\theta\) in the quadratic equation with a function of \(\sin\theta\).

By substituting \(\cos^2\theta\), we convert the entire quadratic equation to involve only \(\sin\theta\). This makes it much easier to handle, as we work with a single trigonometric function rather than two. Trigonometric identities are crucial tools not just for simplifications, but for transforming them into forms that are easier to analyze or solve.

In this problem, we're able to reduce the equation's complexity by focusing on one trigonometric identity, which then leads us to a quadratic inequality in terms of \(\sin\theta\). This kind of substitution helps bridge gaps between algebraic and trigonometric methods, providing a unified approach to the problem.

Solving inequalities is an important aspect of understanding relationships between variables. The exercise involves solving the inequality \(2\sin^2\theta - 3\sin\theta + 1 < 0\), which is a quadratic inequality because the highest degree of \(\sin\theta\) in the expression is two. This can be approached in a similar way to solving quadratic equations, by factoring where possible.

When a quadratic expression can be factored, it often appears as \((2y - 1)(y - 1) < 0\) for variables like \(y = \sin\theta\). The solution to such inequalities involves finding the intervals where the product of these factors is negative.

The key is to solve the inequality step by step:

• First, identify the roots of the factors, \(y = \frac{1}{2}\) and \(y = 1\) in this case.
• Next, use a number line to determine where the product is negative: between these roots, \(\frac{1}{2} < y < 1\).

The logic is the same whether you're working with trigonometric functions or straightforward algebraic expressions. Getting comfortable with factored forms and sign tests is crucial for effectively solving inequalities and interpreting the solutions meaningfully in context.