Question

If \(f(\theta)=(\sin \theta+\operatorname{cosec} \theta)^{2}+(\cos \theta+\sec \theta)^{2}\), then minimum value of \(f(\theta)\) is (1) 7 (2) 8 (3) 9 (4) 4

Short answer

The minimum value of \(f(\theta)\) is 8.

Step-by-step solution

Express given function

Given function is \(f(\theta) = (\sin \theta + \csc \theta)^2 + (\cos \theta + \sec \theta)^2\). Rewrite it for further simplification.

Use trigonometric identities

Recall that \(\csc \theta = \frac{1}{\sin \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\). Therefore, \( \sin \theta + \csc \theta = \sin \theta + \frac{1}{\sin \theta} \) and \( \cos \theta + \sec \theta = \cos \theta + \frac{1}{\cos \theta} \).

Analyze each term separately

Define \(A = \sin \theta + \frac{1}{\sin \theta}\) and \(B = \cos \theta + \frac{1}{\cos \theta}\). Then we have \( f(\theta) = A^2 + B^2 \).

Find minimum value of each term

To find the minimum value of \(A\) and \(B\), use the AM-GM inequality: For any positive numbers \(x\), \(x + \frac{1}{x} \geq 2\). Thus, \( \sin \theta + \frac{1}{\sin \theta} \geq 2 \) and \( \cos \theta + \frac{1}{\cos \theta} \geq 2 \).

Find minimum value of \(A^2 + B^2\)

Since both \(A\) and \(B\) are each greater than or equal to 2, the minimum values of \(A\) and \(B\) each are 2. Hence, \(A = B = 2\). Thus, \( f(\theta) = 2^2 + 2^2 = 4 + 4 = 8\).

Verify the minimum value

Verify that this minimum value can actually be achieved when \(\sin \theta = \cos \theta = \pm \frac{1}{\sqrt{2}} \), meaning \( \theta = \frac{\pi}{4} \) or \( \theta = \frac{3\pi}{4} \). Therefore, the minimum value of \(f(\theta)\) is indeed 8.

Key concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships between trigonometric functions that simplify expressions and solve problems. In this problem, we use identities like \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\) and \(\text{sec} \theta = \frac{1}{\text{cos} \theta}\). These help transform complex trigonometric expressions into simpler ones that are easier to work with. For instance, replacing \(\text{csc} \theta\) and \(\text{sec} \theta\) with their respective identities allows us to express the given function in terms of sinusoidal components.

AM-GM Inequality

The Arithmetic Mean-Geometric Mean (AM-GM) Inequality is a fundamental concept in mathematical optimization. It states that for any set of non-negative numbers, their arithmetic mean is always greater than or equal to their geometric mean. Mathematically, for any positive numbers \[x_1, x_2, \text{...}, x_n\], we have: \[\frac{x_1 + x_2 + \text{...} + x_n}{n} \geq \sqrt[n]{x_1 * x_2 * \text{...} * x_n}\]. In our problem, we apply this inequality to \(\text{sin} \theta + \frac{1}{\text{sin} \theta}\) and \(\text{cos} \theta + \frac{1}{\text{cos} \theta}\), resulting in the conclusion that each expression is greater than or equal to 2, helping us determine the minimum value of the function.

Trigonometric Function Analysis

Analyzing trigonometric functions involves understanding their behavior and properties. For the given function \((\text{sin} \theta + \text{csc} \theta)^2 + (\text{cos} \theta + \text{sec} \theta)^2\), we first simplify by using trigonometric identities. Then we define new expressions \A\ and \B\, which are easier to work with. By studying the bounds of \(\text{sin} \theta\) and \(\text{cos} \theta\), and applying the AM-GM inequality, we can find that both terms have a minimum value of 2 when \(\theta = \frac{\text{π}}{4}\) or \(\theta = \frac{3\text{π}}{4}\). Combining these, we achieve the minimum value of the function.

Mathematical Optimization

Mathematical optimization deals with finding the 'best' solution, such as the minimum or maximum value of a function. In solving \[f(\theta) = (\text{sin} \theta + \text{csc} \theta)^2 + (\text{cos} \theta + \text{sec} \theta)^2\], we seek the minimum value of this function. By analyzing each component using trigonometric identities and inequalities, we identify the minimum value of each part, leading to the overall minimum. Optimization techniques often rely on understanding fundamental properties and applying appropriate mathematical tools like the AM-GM inequality to find the optimal solution efficiently.