Question

The least value of \(2 \sin ^{2} \theta+3 \cos ^{2} \theta\) is : (a) 1 (b) 2 (c) 3 (d) 5

Short answer

Answer: The least value of the trigonometric expression \(2 \sin^2 \theta + 3 \cos^2 \theta\) is 2.

Step-by-step solution

Express \(\sin^2\theta\) in terms of \(\cos^2\theta\)

We know that \(\sin^2 \theta + \cos^2 \theta = 1\). So, we can express \(\sin^2\theta\) as \(\sin^2\theta = 1 - \cos^2\theta\).

Substitute \(\sin^2\theta\) in the given expression

Now, let's substitute the expression for \(\sin^2\theta\) in the given expression: \(2\sin^2\theta + 3\cos^2\theta = 2(1 - \cos^2\theta) + 3\cos^2\theta\)

Simplify the expression

We can simplify the expression as follows: \(2 - 2\cos^2\theta + 3\cos^2\theta = 2 + \cos^2\theta\)

Find the minimum value

Since the minimum value of \(\cos^2\theta\) is 0 (as \(\cos \theta\) varies from -1 to 1), the minimum value of the expression is obtained when \(\cos^2\theta = 0\): \(2 + \cos^2\theta = 2 + 0 = 2\) Thus, the least value of the given expression is 2, which corresponds to the option (b).

Key concepts

Trigonometric Identities

Trigonometric identities are fundamental relationships between the trigonometric functions that hold true for all values of the variable within their domains. These identities help in simplifying expressions and solving trigonometric equations. One of the most commonly used identities is Pythagorean identity, which states:\[\sin^2 \theta + \cos^2 \theta = 1\]This identity can be rewritten in different forms to express one trigonometric function in terms of another. In our exercise, we used it to express \(\sin^2 \theta\) in terms of \(\cos^2 \theta\), which is essential for simplifying the given problem. Mastering these identities is crucial for solving a variety of trigonometric problems.

Optimization Problems

Optimization problems involve finding the maximum or minimum value of a function within a given domain. In trigonometry, this often means finding the extreme values of trigonometric expressions as variables change. These problems are common in engineering, physics, and economics.

• Understand the function or expression to be optimized.
• Determine the range over which the variable can vary.
• Use calculus or algebraic techniques to find the critical points.

In our example, we aimed to find the minimum value of the expression \(2 \sin^2 \theta + 3 \cos^2 \theta\). Knowing that \(\cos^2 \theta\) reaches its minimum at zero helped us quickly identify the condition for which the expression is minimized.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations that represent a particular value or relationship. Simplifying these expressions is a foundational skill in algebra and trigonometry, as it makes it easier to understand and solve complex problems.Consider our expression from the exercise:\[2 \sin^2 \theta + 3 \cos^2 \theta\]Initially, it seems complex, but by substituting \(\sin^2 \theta = 1 - \cos^2 \theta\), we simplified it to \(2 + \cos^2 \theta\). This step made it possible to find the minimum value efficiently.Understanding how to manipulate and simplify expressions allows us to find solutions to mathematical problems more straightforwardly. Practice breaking down expressions and using identities to recognize patterns and simplify your calculations.