Problem 82 - Precalculus Essentials

Question

Use an identity to solve each equation on the interval \([0,2 \pi)\) $$\sin 3 x \cos 2 x+\cos 3 x \sin 2 x=1$$

Step-by-Step Solution

Answer

The solutions to the equation in the interval \([0,2 \pi)\) are \( x = \frac{\pi}{10}, \frac{9\pi}{10}, \frac{17\pi}{10}, \frac{25\pi}{10}, \frac{33\pi}{10} \)

Step 1: Apply the trigonometric identity

Based on the Trigonometric Identity \( \sin(a+b) = \sin a \cos b + \cos a \sin b \), the given equation \( \sin 3x \cos 2x + \cos 3x \sin 2x = 1 \) can be simplified to \( \sin((3x + 2x)) = 1 \), which simplifies further to \( \sin(5x) = 1 \)

Step 2: Solve for x

The solutions to the simplified equation \( \sin(5x) = 1 \) are when \( 5x = \frac{\pi}{2}\) and \(5x = \frac{5\pi}{2}\), so the equation solves to \( x = \frac{\pi}{10}\) and \( x = \frac{\pi}{2}\) respectively. However, the interval is \([0,2 \pi)\), and since \( x = \frac{\pi}{2}\) does not belong to this interval, we will ignore it. To find more solutions for x in the interval, remember that the sine function repeats every \(2 \pi\). So, to find the solutions for x, add multiples of \(\frac{2\pi}{5}\) to the found solution \(x = \frac{\pi}{10}\) until reaching 2π.

Step 3: Find the remaining solutions in the interval

Adding multiples of \(\frac{2\pi}{5}\) to \(x = \frac{\pi}{10}\) will provide the other solutions: \(x = \frac{9\pi}{10}, \frac{17\pi}{10}, \frac{25\pi}{10}, \frac{33\pi}{10} \). All these solutions are within the interval \([0, 2\pi)\).

Key Concepts

Periodic Functions

Periodic functions are those that repeat their values after a certain interval. One of the most familiar periodic functions is the sine function. This means that if you graph the function, it'll repeat a particular pattern over and over again.

• The key aspect of periodic functions is their period, which is the length it takes for the function to complete one full cycle.
• For sine and cosine functions, the period is typically $2\backslash pi$.

In trigonometric equations, you often take advantage of these repeating patterns. For instance, when you solve for solutions within a specified interval, as in the exercise with the function \( \sin(5x) = 1 \), understanding its periodic nature helps us find all possible solutions.By recognizing the periodicity, we can add multiples of the period to find additional solutions within a fixed range. This trick is invaluable for solving equations involving periodic functions.

Sine Function

The sine function is a fundamental trigonometric function that expresses the ratio of the length of the side opposite an angle to the hypotenuse in a right triangle. It features prominently in the study of waves and harmonics due to its periodic properties.

• The sine function is defined for an angle in the context of a unit circle, where the angle is measured from the positive x-axis.
• Mathematically, when dealing with the equation \( \sin(5x) = 1 \), you're looking for angles where the sine value reaches its maximum of 1.

The sine function reaches 1 at \( \frac{\pi}{2} + 2k\pi \), where \( k \) is any integer, due to its periodic nature with period \( 2\pi \).This makes it crucial to know the specific interval in which you are supposed to find your solution. After identifying the first solution, \( x = \frac{\pi}{10} \), additional solutions are found by leveraging the periodicity, incorporating multiples of \( \frac{2\pi}{5} \).

Unit Circle

The unit circle is a circle with a radius of one and centered at the origin of the coordinate plane. It is a crucial concept for understanding trigonometry and its functions.

• Each point on the unit circle has coordinates \( (\cos \theta, \sin \theta) \), where \( \theta \) is an angle measured from the positive x-axis.
• These coordinates directly relate to the trigonometric functions cosine and sine.

When you solve a trigonometric equation like \( \sin(5x) = 1 \), the unit circle helps visualize where this condition is true. Since \( \sin \theta = 1 \) only at \( \theta = \frac{\pi}{2} \) on the unit circle, understanding this relationship enables a better grasp of the concept.By using the unit circle framework, you can clearly determine and predict where the sine and cosine of an angle reach specific values across various intervals.