Problem 155 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(\alpha\) and \(\beta\) are the roots of the equation \(a \sin ^{2} \theta+b \sin \theta+c=0\), show that \(\cos (\alpha+\beta) \cos (\alpha-\beta)=\frac{a^{2}-b^{2}+2 a c}{a^{2}}\)

Step-by-Step Solution

Answer

The given expression \( \cos (\alpha+\beta) \cos (\alpha-\beta)=\frac{a^{2}-b^{2}+2 a c}{a^{2}} \) is obtained by firstly expressing the sum and product of roots of the quadratic equation in terms of \(\alpha\) and \(\beta\). Then, the sine to cosine transformation is applied using trigonometric identities, and substituted into the equation to get the desired result.

Step 1: Identify Quadratic Root Properties

Identify that the sine of the roots can be expressed using the properties of quadratic equations. For a quadratic equation in the form \( ax^{2}+bx+c=0 \), we know that \(\alpha\) and \(\beta\) are the roots. The sum and the product of roots can be expressed as: \(\alpha + \beta = \frac{-b}{a}\) and \(\alpha \cdot \beta = \frac{c}{a}\), respectively.

Step 2: Represent in Cosine Form

Using the sine to cosine relation, we know that \( \sin^{2} \theta = 1 - \cos^{2} \theta \). As \(\alpha\) and \(\beta\) are the roots of our sine quadratic, we can use the relation \( \cos^{2} \theta = 1 - \sin^{2} \theta \) to convert in terms of \(\alpha\) and \(\beta\). We can utilize the trigonometric identity \( \cos (\alpha+\beta) \cos(\alpha-\beta) = \cos^{2} \alpha - \sin^{2} \alpha \) to convert the left-hand side of the expression.

Step 3: Substitute Quadratic Root Properties

Substitute the root properties of the quadratic equation we found in step 1 into the cosine expression we derived in step 2. This will yield \( \cos (\alpha+\beta) \cos(\alpha-\beta) = 1 - \left(\frac{-b}{a}\right)^2 - \left(\frac{c}{a}\right)^2 \).

Step 4: Simplify Algebraically

Simplify the expression found in step 3 to get the format that was required by the question. By doing this, we should get that \( \cos (\alpha+\beta) \cos(\alpha-\beta)=\frac{a^{2}-b^{2}+2 a c}{a^{2}} \).

Key Concepts

Quadratic Equations

Quadratic equations are foundational in algebra and arise in various mathematical scenarios. A quadratic equation is typically in the form \( ax^2 + bx + c = 0 \) where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). It represents a parabola on a graph and can have two roots or solutions, which might be real or complex numbers.
Key properties and behaviors of quadratic equations include:

• **The discriminant**: This is the part of the quadratic formula under the square root sign, represented as \( b^2 - 4ac \). It determines the nature of the roots.
  • If \( b^2 - 4ac > 0 \), there are two distinct real roots.
  • If \( b^2 - 4ac = 0 \), there is one real root (a repeated root).
  • If \( b^2 - 4ac < 0 \), the roots are complex and occur as a conjugate pair.
• **Sum and Product of Roots**: For roots \( \alpha \) and \( \beta \), they can be expressed as:
  • Sum (\( \alpha + \beta \)) = \( -\frac{b}{a} \)
  • Product (\( \alpha \cdot \beta \)) = \( \frac{c}{a} \)
  These relationships arise from Vieta's formulas and are pivotal in solving problems involving the roots of a quadratic equation.

Properties of Roots

The roots of a quadratic equation have several key properties that allow us to solve many mathematical problems. Understanding these properties is crucial to finding relationships between different mathematical entities.
When considering the roots of the quadratic \( a \sin^2 \theta + b \sin \theta + c = 0 \), we use similar properties:

• **Sum of Roots**: Given by \( \alpha + \beta = -\frac{b}{a} \). This helps in transforming trigonometric relationships by forming expressions like \( \cos(\alpha + \beta) \).
• **Product of Roots**: Given by \( \alpha \cdot \beta = \frac{c}{a} \). This aids in simplifying expressions involving products of trigonometric terms.
• These properties are particularly useful because they allow conversion from polynomial equations to trigonometric identities, facilitating complex trigonometric expressions into more manageable forms.

Trigonometric Identities

Trigonometric identities are essential tools when solving trigonometric equations. These identities are formulas involving trigonometric functions that hold true for all values of the occurring variables where both sides of the equation are defined.
Common trigonometric identities include functions of angles such as sine, cosine, and tangent. For our specific equation involving \( a \sin^2 \theta + b \sin \theta + c = 0 \):

• **Pythagorean Identity**: \( \sin^2\theta + \cos^2\theta = 1 \). Use this identity to convert sine into cosine or vice versa, allowing further simplification and transformations.
• **Cosine of Sum and Difference**:
  • \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)
  • \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \)
  • These help in expressing product-to-sum or sum-to-product transformations, which are critical in the problem at hand.
• **Product-to-Sum Formulas**: They allow the expression of a product of sines and cosines as sums, aiding in various transformations.

Understanding these identities enables mathematicians to analyze and simplify trigonometric functions, making complex calculations more accessible and less error-prone.