Problem 13 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Find the maximum or minimum values of: (i) \(3 \cos x+4 \sin x+8\) (ii) \(5 \cos x+3 \cos \left(x+\frac{\pi}{3}\right)+3\)

Step-by-Step Solution

Answer

(i) The function \(3 \cos x+4 \sin x+8\) has its maximum value at \(r + 8\) and its minimum value at \(8 - r\). \n\n(ii) The function \(5 \cos x+3 \cos \left(x+\frac{\pi}{3}\right)+3\) has its maximum value at \(a + b + 3\) and its minimum value at \(3 - (a + b)\).

Step 1: Rewrite each function using an auxiliary angle

(i) Rewrite the function \(3 \cos x+4 \sin x+8\) using the auxiliary angle \(\tan \phi=\frac{3}{4}\), to get it in the form \(r \cos (x - \phi)+ 8\), where \(r\) is the magnitude of the function. \n\n(ii) For the function \(5 \cos x+3 \cos \left(x+\frac{\pi}{3}\right)+3\), it's not trivial to directly rewrite it into a single cosine function, so split it into two parts: \(5 \cos x\) and \(3 \cos \left(x+\frac{\pi}{3}\right)\). Both can be rewritten with an auxiliary angle to get the forms \(a \cos (x - \alpha) + b \cos (x - \beta) + 3\).

Step 2: Determine the maximum and minimum of each function

(i) \(r \cos (x - \phi) + 8\) will maximise when the cosine function is maximum, i.e at \(\cos (x - \phi)=1\). Hence, the maximum value of the function is \(r + 8\) and its minimum value is \(8 - r\). \n\n(ii) For each cosine component in the function, the maximum occurs at \(\cos (x - \alpha)=1\) and \(\cos (x - \beta)=1\). Hence, the maximum value of \(a \cos (x - \alpha) + b \cos (x - \beta) + 3\) is \(a + b + 3\); and its minimum value is \(3 - (a + b)\).

Key Concepts

Auxiliary Angle Method

The Auxiliary Angle Method is a technique used in trigonometry to simplify expressions combining sine and cosine functions into a single trigonometric function. This is particularly useful in finding maximum and minimum values. Here's how it works:

• The key idea is to express combined sine and cosine terms in the form of a single term using an angle shift. This form is visually simpler and makes it easier to identify extrema.
• For an expression like \(a \cos x + b \sin x\), we aim to reframe it as \(r \cos (x - \phi)\), where \(r\) is the amplitude, and \(\phi\) is the phase angle.
• To find \(r\), use the formula: \(r = \sqrt{a^2 + b^2}\).
• The phase angle \(\phi\) can be determined by \(\tan \phi = \frac{b}{a}\).
• This method is convenient as it compresses the expression into a simple cosine function, which we can easily manipulate to find extremes.

In our example, the expression \(3 \cos x + 4 \sin x + 8\) uses this method to form \(r \cos (x - \phi) + 8\). By doing so, evaluating maximum and minimum values becomes straightforward.

Maximum and Minimum Values

Determining the maximum and minimum values of trigonometric functions is a common problem in trigonometry. Once an expression is simplified using the Auxiliary Angle Method, finding these values is more straightforward.

• For a transformed trigonometric function like \(r \cos (x - \phi) + c\), where \(r\) is the magnitude or amplitude and \(c\) is a constant, the value of the function varies between \(c - r\) and \(c + r\).
• The cosine function achieves its maximum value of 1 and its minimum value of -1. So, in the given transformed function, \(r \cos (x - \phi) + c\), the maximum occurs when \(\cos (x - \phi) = 1\), making the maximum value \(r + c\).
• Similarly, the minimum occurs when \(\cos (x - \phi) = -1\), producing the minimum value \(c - r\).

In the examples given, the problem illustrates this process by simplifying the trigonometric forms using auxiliary angles, allowing us to immediately calculate these extreme values.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, characterized by its periodic wave pattern. An understanding of its properties is essential for solving trigonometric problems.

• The function \(\cos x\) oscillates between -1 and 1, with a periodicity of \(2\pi\), which means it repeats every \(2\pi\) units.
• It achieves its maximum value of 1 at angles like \(0, 2\pi, 4\pi,\) etc., and its minimum value of -1 at \(\pi, 3\pi, 5\pi,\) etc.
• By understanding the behavior of the cosine function, you can predict its value change in transformed functions like \(a \cos (x - \phi) + c\).
• In these transformed functions, the shift \(x - \phi\) indicates a phase shift in the wave, altering when the maxima and minima occur without changing their values.

In the problems at hand, this fundamental knowledge of the cosine function aids in determining how transformations using auxiliary angles impact its maxima and minima in complex trigonometric expressions.