Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ 4 \sin ^{2} x-3=0 $$

Step-by-Step Solution

Answer

The solutions are \( \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3} \) in the interval \([0, 2\pi)\).

Step 1: Isolate the Trigonometric Function

Begin by isolating the squared sine function. Start with the equation:\[ 4\sin^{2}x - 3 = 0 \]Add 3 to both sides:\[ 4\sin^{2}x = 3 \]Now, divide both sides by 4:\[ \sin^{2}x = \frac{3}{4} \]

Step 2: Take the Square Root

To find \( \sin x \), take the square root of both sides. Remember that taking the square root involves both the positive and negative solutions:\[ \sin x = \pm \sqrt{\frac{3}{4}} \]Simplify the square root:\[ \sin x = \pm \frac{\sqrt{3}}{2} \]

Step 3: Determine Possible Angles

Now, we need to find all angles \( x \) in the interval \([0, 2\pi)\) where \( \sin x = \frac{\sqrt{3}}{2} \) and \( \sin x = -\frac{\sqrt{3}}{2} \).For \( \sin x = \frac{\sqrt{3}}{2} \), the solutions in the standard interval are:\[ x = \frac{\pi}{3}, \frac{2\pi}{3} \].For \( \sin x = -\frac{\sqrt{3}}{2} \), the solutions are:\[ x = \frac{4\pi}{3}, \frac{5\pi}{3} \].

Step 4: Write the General Solution

Since the sine function is periodic with period \(2\pi\), we include the general solutions:- \( x = \frac{\pi}{3} + 2k\pi \)- \( x = \frac{2\pi}{3} + 2k\pi \)- \( x = \frac{4\pi}{3} + 2k\pi \)- \( x = \frac{5\pi}{3} + 2k\pi \)for any integer \(k\).

Key Concepts

Periodic Functions

Periodic functions are an essential part of trigonometry and mathematics as a whole. They repeat their values in regular intervals or periods. For example, the sine function is periodic, meaning that it repeats its values every specific interval known as the period. In mathematical terms, a function \( f(x) \) is called periodic if there exists a non-zero constant \( P \) such that:

• \( f(x + P) = f(x) \)

This characteristic is what makes it possible to predict the behavior of the function over its cycle. The most common period for sine and cosine functions is \( 2\pi \).
Therefore, understanding periodicity is crucial for solving trigonometric equations, especially when seeking all possible solutions. This is often applied in real-world scenarios like sound waves or seasonal patterns where repetition is evident.

Sine Function

The sine function (\( \sin x \)) is a fundamental trigonometric function. It represents one of the corners of understanding periodic behavior in mathematics. The sine function fluctuates between -1 and 1 as it moves along its period. Its graph forms a smooth wave, known as the sine wave.
In our exercise, we dealt with the equation \( \sin^2 x = \frac{3}{4} \). By taking the square root, we explored the possible solutions \( \sin x = \pm \frac{\sqrt{3}}{2} \). These values correspond to angles on the unit circle where the sine value reaches \( \frac{\sqrt{3}}{2} \) and \(-\frac{\sqrt{3}}{2} \).

• Positive \( \sin x \) points refer to angles in the first and second quadrants.
• Negative \( \sin x \) points to angles in the third and fourth quadrants.

Comprehending the unit circle and its quadrants is imperative when determining related angles. This allows learners to source the complete set of solutions for sine-related equations.

General Solution

When solving trigonometric equations, it's important to find what's known as the general solution. This refers to the entire set of solutions, not just those that are initially discovered.Given the periodic nature of trigonometric functions like sine, these solutions repeat every \( 2\pi \). This is vital to know since initial solutions are just the starting points.
In the solved problem, the general solutions include:

• \( x = \frac{\pi}{3} + 2k\pi \)
• \( x = \frac{2\pi}{3} + 2k\pi \)
• \( x = \frac{4\pi}{3} + 2k\pi \)
• \( x = \frac{5\pi}{3} + 2k\pi \)

Here, \( k \) is any integer, representing the multiples of the period. This leads to all possible angles where the sine has the given values. Thus, when tackling trigonometric problems, always link the solutions by adding or subtracting periods for a comprehensive solution.