Trigonometric Equations: Problems based on Extreme Values

Trigonometric equations are mathematical statements that involve trigonometric functions such as sine, cosine, tangent, etc., and the solutions to these equations are the angles that satisfy the given equation. In the context of extreme values, we are often interested in finding the maximum or minimum values that a trigonometric expression can take. This is particularly useful in various fields such as physics, engineering, and optimization problems.

Understanding Extreme Values

Extreme values refer to the highest or lowest points in a given domain. In trigonometry, these values are often found by analyzing the behavior of the trigonometric functions.

Important Points:

• Periodicity: Trigonometric functions are periodic, which means they repeat their values in regular intervals. For example, the sine and cosine functions have a period of $2\pi$.
• Amplitude: The amplitude of a trigonometric function is the maximum absolute value it can take. For the basic sine and cosine functions, the amplitude is 1.
• Phase Shift: A horizontal shift in the graph of a trigonometric function is called a phase shift.
• Frequency: The number of cycles a trigonometric function completes in a unit interval.

Table of Basic Trigonometric Functions and Their Extreme Values

Function	Period	Amplitude	Domain	Range	Extreme Values
$\sin(x)$	$2\pi$	1	$(-\infty, \infty)$	$[-1, 1]$	Max: $\sin(x) = 1$ when $x = \frac{\pi}{2} + 2k\pi$ Min: $\sin(x) = -1$ when $x = \frac{3\pi}{2} + 2k\pi$
$\cos(x)$	$2\pi$	1	$(-\infty, \infty)$	$[-1, 1]$	Max: $\cos(x) = 1$ when $x = 2k\pi$ Min: $\cos(x) = -1$ when $x = \pi + 2k\pi$
$\tan(x)$	$\pi$	$\infty$	$x \neq \frac{\pi}{2} + k\pi$	$(-\infty, \infty)$	No finite extreme values

Note: $k$ is an integer representing the number of periods.

Solving Trigonometric Equations for Extreme Values

To solve trigonometric equations for extreme values, we often use differentiation if the function is differentiable. For basic trigonometric functions, we can also use their known properties and graphs.

Example 1: Find the extreme values of $f(x) = \sin(x) + \cos(x)$

To find the extreme values of $f(x)$, we can use the following steps:

1. Find the derivative of $f(x)$: $f'(x) = \cos(x) - \sin(x)$
2. Set the derivative equal to zero: $\cos(x) - \sin(x) = 0$
3. Solve for $x$: $\tan(x) = 1 \Rightarrow x = \frac{\pi}{4} + k\pi$
4. Determine if these are maxima or minima: Second derivative test or analyze the sign change of $f'(x)$ around the critical points.

Example 2: Maximize $f(x) = \sin^2(x)$ for $x \in [0, 2\pi]$

1. Find the derivative of $f(x)$: $f'(x) = 2\sin(x)\cos(x)$
2. Set the derivative equal to zero: $2\sin(x)\cos(x) = 0$
3. Solve for $x$: $x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$
4. Evaluate $f(x)$ at these points: $f(0) = f(\pi) = f(2\pi) = 0$, $f\left(\frac{\pi}{2}\right) = f\left(\frac{3\pi}{2}\right) = 1$
5. Determine the maximum: The maximum value is $1$ at $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$.

Conclusion

Problems based on extreme values in trigonometric equations require a good understanding of the properties of trigonometric functions and the ability to apply calculus techniques such as differentiation. By following a systematic approach, one can find the maximum and minimum values of trigonometric expressions, which is a valuable skill in various applications.