Problem 110 - Algebra and Trigonometry

Question

Solve: $$\text 2 \sin ^{2} x-1=0,0 \leq x<2 \pi$$

Step-by-Step Solution

Answer

The solution to the equation \(2 \sin ^{2} x-1=0\) for \(0 \leq x<2 \pi\) is \(x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\).

Step 1: Move Constants to Right Side

The equation given is \(2 \sin^2 x - 1 = 0\). In this step, move -1 to the other side of the equation to get: \(2 \sin^2 x = 1\).

Step 2: Isolate Sin Squared

Now to isolate \(\sin^2 x\), divide both sides of the equation by 2. This gives: \(\sin^2 x = \frac{1}{2}\).

Step 3: Square Root Both Sides

Take the square root of both sides to isolate \(\sin x\). Remember, when taking the square root of a squared function, to include the \(\pm\) solution. This then gives: \(\sin x = \pm \sqrt{\frac{1}{2}} = \pm \frac{1}{\sqrt{2}}\).

Step 4: Determine Possible x Values

Given the values of sine equals \(\pm \frac{1}{\sqrt{2}}\), the angles for \(x\) in the range [0, 2\(\pi\)) can be determined to be \(x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\)

Key Concepts

Sin^2 x Trigonometric Identity

Trigonometric identities are equations that hold true for any value of the variable within the domain. When dealing with trigonometric equations, knowing these identities can simplify the problem-solving process. In particular, the identity involving \(\sin^2 x\) is one of the Pythagorean identities, which states that \(\sin^2 x + \cos^2 x = 1\).

When solving trigonometric equations, it is common to manipulate the original equation to use these identities effectively. For example, in the given exercise, you can rewrite \(\sin^2 x\) as \(1 - \cos^2 x\) to apply the Pythagorean identity. This strategic use of trigonometric identities is crucial in simplifying complex trigonometric equations and making them solvable.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find the angle that corresponds to a given trigonometric ratio. For instance, if we know the value of \(\sin x\), we can use the inverse function \(\sin^{-1}\) to find the angle \(x\). These functions are denoted as \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\), among others.

In our exercise, we find that \(\sin x = \pm \frac{1}{\sqrt{2}}\). To determine the angles that satisfy this equation, we would typically use the inverse sine function. However, this exercise requires knowledge of the unit circle and the specific angles for which the sine value is \(\pm \frac{1}{\sqrt{2}}\) rather than using the inverse function directly. Knowing these angles beforehand is a faster approach in this context.

Trigonometry Problem Solving

When solving trigonometry problems, it's essential to follow a structured approach and understand the relationships between trigonometric functions and angles. The steps usually include isolating the trigonometric function, using identities to simplify the equation, and then finding the angles that satisfy the conditions of the problem.

In our exercise, we isolated \(\sin x\) by taking a square root and then looked for angles that satisfy \(\sin x = \pm \frac{1}{\sqrt{2}}\) in the specified interval \(0 \leq x < 2\pi\). By doing so, we reduce trigonometric equations to a form that is straightforward to solve. Practice in recognizing patterns in these equations and fluency with trigonometric identities greatly assists in effective problem solving.

Unit Circle Angles

The unit circle is a fundamental concept in trigonometry, representing all the angles and associated trigonometric values on a circle with a radius of one. By using the unit circle, we can find the sine, cosine, and tangent values of angles that are often used in trigonometry problems.

In context with our exercise, we must identify the angles within the range \(0 \leq x < 2\pi\) where the \(\sin x\) is \(\pm \frac{1}{\sqrt{2}}\). These angles correspond to points on the unit circle where the y-coordinate (which represents the\sin value) is \(\pm \frac{1}{\sqrt{2}}\). It is here where the unit circle becomes a powerful tool, allowing us to quickly identify \(x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\) as the angles that meet the criteria.