Problem 76 - Precalculus Essentials

Question

Use an identity to solve each equation on the interval \([0,2 \pi)\) $$\sin x \cos x=\frac{\sqrt{3}}{4}$$

Step-by-Step Solution

Answer

So the solutions for the equation \(\sin x \cos x = \frac{\sqrt{3}}{4}\) within the interval \([0, 2\pi)\) are \(x = \frac{\pi}{6}\), \(x = \frac{\pi}{3}\), \(x = \frac{7\pi}{3}\), and \(x = \frac{13\pi}{6}\).

Step 1: Transform the Equation Using Trigonometric Identity

Use the identity \(2 \sin x \cos x = \sin 2x\). Our equation then becomes \(\sin 2x = \frac{\sqrt{3}}{2}\) by multiplying the original equation by 2.

Step 2: Determine Solutions From Unit Circle

Now, we need to find the values of \(2x\) that make the equation true. Looking at the unit circle, \(\sin \theta = \frac{\sqrt{3}}{2}\) at \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{2\pi}{3}\). Thus, \(2x = \frac{\pi}{3}\) or \(2x = \frac{2\pi}{3}\).

Step 3: Solve for x

Solving for 'x' from the equations \(2x = \frac{\pi}{3}\) and \(2x = \frac{2\pi}{3}\), we get \(x = \frac{\pi}{6}\) or \(x = \frac{\pi}{3}\). Since the range is from 0 to \(2\pi\), we need to also include the coterminal angles for these solutions.

Step 4: Find the Coterminal Angles

To find the coterminal angles, add \(2\pi\) to each solution. So for \(x = \frac{\pi}{6}\), the coterminal angle is \(\frac{\pi}{6} + 2\pi = \frac{13\pi}{6}\). For \(x = \frac{\pi}{3}\), the coterminal angle is \(\frac{\pi}{3} + 2\pi = \frac{7\pi}{3}\). Both these angles are in the given interval of \([0, 2\pi)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in solving trigonometric equations. They allow us to rewrite trigonometric functions in various forms. For instance, the identity used in our exercise, \(2 \sin x \cos x = \sin 2x\), is known as the double angle identity for sine. This is particularly useful because it simplifies the equation, making it easier to solve.

There are numerous trigonometric identities, including Pythagorean identities, sum and difference formulas, and half-angle identities, each serving a unique purpose. Understanding when and how to apply these identities is crucial for solving trigonometric equations effectively.

It's important to get acquainted with these identities, as they are commonly used to transform equations into a more solvable form. For any student, memorizing the basic identities and practicing their application can significantly streamline the process of solving trigonometric problems.

Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of a coordinate system. It is particularly useful in trigonometry for visualizing and defining sine, cosine, and tangent values for all angles. Each point on the unit circle represents an angle's cosine and sine as the x and y coordinates, respectively.

For our exercise, we used the unit circle to find the angles where the sine value equals \(\frac{\sqrt{3}}{2}\). Knowing that the sine function corresponds to the y-coordinate in the unit circle, we could determine that the possible angles are \(\frac{\pi}{3}\) and \(\frac{2\pi}{3}\). These angles correspond to the standard positions where the sine function has the desired value.

Mastering the unit circle can be highly beneficial, as it not only helps in solving equations but also provide a deeper understanding of trigonometric functions and their relationships with angles.

Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides, but may differ by any number of full rotations. In terms of our exercise, we find coterminal angles by adding or subtracting \(2\pi\) (or 360 degrees in the degree measure) to the primary angles found from solving the equation.

In practical terms, if you have an angle \(\theta\), any angle \(\theta + 2k\pi\), where \(k\) is an integer, is coterminal with \(\theta\). This is because the angles end at the same position after a full rotation or multiple rotations of the unit circle.

When solving trigonometric equations, especially in a specified interval, it’s crucial to consider coterminal angles because they can provide additional solutions within the given range. As in step 4 of our exercise, we found coterminal angles to ensure we captured all solutions within the interval \([0, 2\pi)\). Recognizing and calculating coterminal angles are essential skills in trigonometry and can be widely applied in geometrical problems and periodic functions.