Problem 10 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Show that for all values of \(\theta\), the expression \(a \sin ^{2} \theta+b \sin \theta \cos \theta+c \cos ^{2} \theta\) lies between \(\frac{a+c}{2}-\frac{\sqrt{b^{2}+(a-c)^{2}}}{2}\) and \(\frac{a+c}{2}+\frac{\sqrt{b^{2}+(a-c)^{2}}}{2}\)

Step-by-Step Solution

Answer

The expression \(a\sin^2\theta + b\sin\theta\cos\theta + c\cos^2\theta\) lies between \(\frac{a+c}{2}-\frac{\sqrt{b^{2}+(a-c)^{2}}}{2}\) and \(\frac{a+c}{2}+\frac{\sqrt{b^{2}+(a-c)^{2}}}{2}\), for all values of \(\theta\). This is confirmed by manipulating the given expression with the use of trigonometric identities and the properties of quadratic functions.

Step 1: Write Down Given Expression

The given expression is \(a\sin^2\theta + b\sin\theta\cos\theta + c\cos^2\theta\). We need to find the range (min and max) of this expression.

Step 2: Recall and Apply Trigonometric Identities

Use the trigonometric identity \(\sin^2\theta+\cos^2\theta=1\). Notice that, if \(x=\sin\theta\), then \(1-x^2\) will be \(\cos^2\theta\). So, let us substitute this into the expression and start breaking it down to its minimum and maximum. That is, rewrite the expression as \(ax^2 + bxc + c(1-x^2)\). After simplifying, this yields \((a-c)x^2 + bxc + c\). This is a quadratic equation in \(x\).

Step 3: Determine Extremes of Quadratic Function

Using the knowledge of quadratic functions, we know that for a quadratic function in the form of \(ax^2 + bx + c\), the minimum or maximum is given by \(-b/(2a)\) if \(a\) is not equal to zero. If \(a = c\), then the expression is a linear function of \(\sin\theta\) and the minimum and maximum are achieved when \(\sin\theta\) equals -1 and 1 respectively. The minimum and maximum of \(ax^2 + bxc + c\) for \(a \neq c\) are attained when \(x=\sin\theta = -b/(2(a-c))\), if \(-1 \leq -b/(2(a-c)) \leq 1\) or at \(x = \sin\theta = \pm 1\) otherwise.

Step 4: Calculate the Maximum and Minimum Values

Calculate the maximum and minimum of the given expression using these values of \(\sin\theta\). After substituting and simplifying, you obtain the respective expressions for the maximum and the minimum. The maximum is \(\frac{a+c}{2}+\frac{\sqrt{b^{2}+(a-c)^{2}}}{2}\); the minimum is \(\frac{a+c}{2}-\frac{\sqrt{b^{2}+(a-c)^{2}}}{2}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry that relate the trigonometric functions to one another. In this exercise, one of the most important identities used is the Pythagorean identity, which states that for any angle \( \theta \), \( \sin^2\theta + \cos^2\theta = 1 \). This identity is crucial in simplifying trigonometric expressions by allowing the substitution of one trigonometric function in terms of another.

• By letting \( x = \sin\theta \), we automatically know that \( \cos^2\theta = 1 - x^2 \).
• This substitution transforms the original trigonometric expression into a quadratic structure in terms of \( x \).

Having a good grasp of these basic identities helps in expressing complex trigonometric equations in simpler forms. These forms can further be solved with algebraic methods, providing a bridge between trigonometry and algebra.

Quadratic Functions

Quadratic functions take the general form of \( ax^2 + bx + c \) and have a characteristic parabolic shape when graphed. These functions can open upwards or downwards depending on the sign of the coefficient \( a \). In the context of this problem, a transformed version of the original trigonometric expression is compared to a quadratic in the variable \( x = \sin\theta \).

• The expression \( (a-c)x^2 + bxc + c \) is recognized as a quadratic function in the form of \( ax^2 + bx + c \).
• The calculation of the vertex of this parabola provides the key for finding the expression's extreme values.

For the minimum and maximum values, the vertex formula \( -b/(2a) \) is used to find where these extremities are located along the x-axis, providing critical boundaries for the values of \( \theta \). Understanding this property of quadratics aids in efficiently solving the original trigonometric problem.

Expression Minimization

Expression minimization involves finding the smallest value that an expression can take by mathematical manipulation and optimization methods. In this exercise, minimization is carried out by analyzing the quadratic form derived from the trigonometric expression. The strategy employed here taps into the properties of quadratic functions to determine potential minimum (and maximum) values.

• Identify critical points using \( x = \sin\theta = -b/(2(a-c)) \) when \( -1 \leq x \leq 1 \).
• If this calculated \( x \) falls outside \([-1, 1]\), then check at \( x = \pm 1 \), corresponding to \( \sin\theta = 1 \) or \( \sin\theta = -1 \).
• Evaluate the expression at these critical and boundary points to determine the smallest value it can achieve.

Through careful calculation, the determined minimum and maximum expressions verify that the solution is enclosed within the specified range, showcasing the power of combining algebra with trigonometry in minimizing expressions.