Finding the Relationship between Coefficients and Roots

The problem asks for the relationship between the coefficients \(a\), \(b\), and \(c\) of a quadratic equation \(ax^2 + bx + c = 0\), given that its roots are \(\sin \theta\) and \(\cos \theta\).

For any quadratic equation in the form \(Ax^2 + Bx + C = 0\), the relationship between the roots (let's call them \(\alpha\) and \(\beta\)) and the coefficients is given by Vieta's formulas:

• Sum of roots: \(\alpha + \beta = -\frac{B}{A}\)
• Product of roots: \(\alpha \cdot \beta = \frac{C}{A}\)

In our given equation, \(ax^2 + bx + c = 0\), the coefficients are \(A=a\), \(B=b\), and \(C=c\). The roots are \(\sin \theta\) and \(\cos \theta\). Applying Vieta's formulas to this equation:

• Sum of roots: \(\sin \theta + \cos \theta = -\frac{b}{a}\)
• Product of roots: \(\sin \theta \cdot \cos \theta = \frac{c}{a}\)

We also know a fundamental trigonometric identity:

• \(\sin^2 \theta + \cos^2 \theta = 1\)

We can use the sum of roots equation and the trigonometric identity to find the required relationship. Let's square the sum of roots equation:

\((\sin \theta + \cos \theta)^2 = \left(-\frac{b}{a}\right)^2\)

Expand the left side using the identity \((x+y)^2 = x^2 + y^2 + 2xy\):

\(\sin^2 \theta + \cos^2 \theta + 2 \sin \theta \cos \theta = \frac{b^2}{a^2}\)

Now, substitute the trigonometric identity \(\sin^2 \theta + \cos^2 \theta = 1\) into this equation:

\(1 + 2 \sin \theta \cos \theta = \frac{b^2}{a^2}\)

Next, substitute the product of roots \(\sin \theta \cos \theta = \frac{c}{a}\) into the equation:

\(1 + 2 \left(\frac{c}{a}\right) = \frac{b^2}{a^2}\)

Simplify the equation:

\(1 + \frac{2c}{a} = \frac{b^2}{a^2}\)

To eliminate the denominators, multiply the entire equation by \(a^2\) (assuming \(a \neq 0\), which must be true for a quadratic equation):

\(a^2 \cdot 1 + a^2 \cdot \frac{2c}{a} = a^2 \cdot \frac{b^2}{a^2}\)

\(a^2 + 2ac = b^2\)

Rearrange the terms to match the format of the given options:

\(a^2 - b^2 + 2ac = 0\)

This derived equation is the relationship between \(a\), \(b\), and \(c\).

Let's compare this with the given options:

• Option 1: \(a^2 + b^2 - 2ac = 0\)
• Option 2: \(-a^2 + b^2 + 2ac = 0\)
• Option 3: \(a^2 - b^2 + 2ac = 0\)
• Option 4: \(a^2 + b^2 + 2ac = 0\)

Our derived relationship \(a^2 - b^2 + 2ac = 0\) matches Option 3.

Revision Table: Key Concepts

Additional Information: Understanding Roots and Coefficients

The connection between the roots and coefficients of a polynomial is a powerful tool in algebra. Vieta's formulas allow us to deduce properties of the roots without actually solving the equation. In this problem, by knowing the specific nature of the roots (being sine and cosine of the same angle \(\theta\)), we could use a trigonometric identity (\(\sin^2 \theta + \cos^2 \theta = 1\)) to establish a constraint on the coefficients \(a\), \(b\), and \(c\).

The process involved squaring the sum of roots because the sum \(\sin \theta + \cos \theta\) when squared, conveniently includes the term \(\sin^2 \theta + \cos^2 \theta\) which simplifies to 1, and the term \(2\sin \theta \cos \theta\) which is twice the product of the roots. This transformation is key to connecting the algebraic properties (sum and product of roots) with the trigonometric properties of the specific roots given.