Problem 24 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. The minimum value of \(2^{\sin x}+2^{\cos x}\) is (a) 1 (b) \(2^{1-\frac{1}{\sqrt{2}}}\) (c) \(2^{-\frac{1}{\sqrt{2}}}\) (d) \(\left(2-\frac{1}{\sqrt{2}}\right)\)

Step-by-Step Solution

Answer

The correct answer is (a) 1.

Step 1: Determine the Range of Sin(x) and Cos(x)

Determine the maximum and minimum value of sine and cosine since they are the argument to the exponents. The sine function and the cosine function both vary in the interval \([-1,1]\). Therefore, the smallest value they can take is -1 and the largest value that they can take is 1.

Step 2: Determine the minimum value of the function

The function \(2^x\) is an increasing function, therefore, it takes its lowest value when \(x\) is at its smallest. In this case, when \(\sin(x)=-1\) or \(\cos(x)=-1\) and takes the largest value when \(\sin(x)=1\) or \(\cos(x)=1\). Therefore, the function \(2^{\sin x}+2^{\cos x}\) will be at its minimum when both \(\sin x\) and \(\cos x\) are equal to -1 and at its maximum when both are equal to 1.

Step 3: Calculating the minimum value

When \(\sin x = -1\) and \(\cos x = -1\), the function becomes \(2^{-1} + 2^{-1}\) which simplifies to \(\frac{1}{2} + \frac{1}{2} = 1\). This is the minimum value of the function.

Key Concepts

Minimum Value of Function

To find the minimum value of a function, we begin by exploring its behavior based on its input values. This involves considering the range and behavior of functions inputted to, in this case, the exponential function. The function we are considering is \(2^{\sin x} + 2^{\cos x}\). Our goal is to determine the minimum value this function can achieve.

To understand this better, consider the individual components: \(2^{\sin x}\) and \(2^{\cos x}\). Since the exponential function \(2^x\) is an increasing function, it achieves its smallest value when \(x\) is at its smallest.

In our scenario, the function takes its minimum when both \(\sin x\) and \(\cos x\) are smallest. From the range of these trigonometric functions, these smallest values occur at -1. Hence, substituting we get, \(2^{-1} + 2^{-1} = \frac{1}{2} + \frac{1}{2} = 1\). This confirms that the minimum value of the function \(2^{\sin x} + 2^{\cos x}\) is 1.

Sine and Cosine Range

The sine and cosine functions are fundamental components of trigonometry. Understanding their ranges is crucial for solving problems involving these functions.

Both \(\sin x\) and \(\cos x\) functions oscillate between -1 and 1. This means that for any angle \(x\), the sine and cosine of that angle will lie within the interval \([-1, 1]\).

• This range is determined by the unit circle on the Cartesian plane, where both functions measure projected lengths, leading them to repeat values in cycles from -1 to 1.
• This cyclical nature means that regardless of the input angle \(x\), the output will always fall within this fixed range.

Understanding this range helps dictate the possible outputs that functions containing \(\sin x\) and \(\cos x\) can have, which in turn affects calculations like finding minimum or maximum values in more complex functions like \(2^{\sin x} + 2^{\cos x}\).

Exponential Function

Exponential functions are a key tool in mathematics, known for their rapid growth and specific properties. In the function \(2^x\), you find that for any positive base greater than one, such as 2, the function is continuous and strictly increasing.

• This characteristic of being **increasing** means that when the input, \(x\), increases, the output value of \(2^x\) always increases, and conversely, when \(x\) decreases, \(2^x\) also decreases.
• Thus, to achieve the smallest value of an exponential function like \(2^{\sin x}\) or \(2^{\cos x}\), you want \(\sin x\) or \(\cos x\) to be at their smallest possible value based on their known range.

This behavior of exponential functions is pivotal in determining and understanding the outputs when combined with other functions, such as those involving trigonometric inputs. It's the predictable increase or decrease that allows for strategic calculation of minimum and maximum possible values.