Problem 7 - Algebra and Trigonometry Real Mathematics, Real People

Question

Value is a solution of the equation. \(3 \tan ^{2} 2 x-1=0\) (a) \(x=\frac{\pi}{12}\) (b) \(x=\frac{5 \pi}{12}\)

Step-by-Step Solution

Answer

(a) and (b) are both the correct answer. The values \(x=\frac{\pi}{12}\) and \(x=\frac{5 \pi}{12}\) are both solutions to the given equation \(3 \tan ^{2} 2 x-1=0\).

Step 1: Simplify the given equation

Rewrite the equation \(3 \tan ^{2} 2 x-1=0\) as \(\tan ^{2} 2 x = \frac {1}{3}\). This corresponds to \(\tan 2x = \pm \sqrt{\frac {1}{3}} = \pm \frac{1}{\sqrt{3}}\). The solutions to this function will be \(2x = \arctan (\pm \frac{1}{\sqrt{3}})\) which results in \(2x = \frac{\pi}{6}, \frac{5\pi}{6}\). So, the possible solutions for x are \(x = \frac{\pi}{12}, \frac{5\pi}{12}\).

Step 2: Test possible solutions

We now test both possible solutions. Using (a) \(x = \frac{\pi}{12}\), we get \(2x = \frac{\pi}{6}\) which is a solution. Testing (b) \(x=\frac{5\pi}{12}\), we have \(2x = \frac{5\pi}{6}\) and this is also a solution to our simplified equation.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential mathematical tools for working with angles and their relationships. They include the sine, cosine, tangent, and their inverse functions. Understanding these functions is pivotal when solving trigonometric equations like the one given in the exercise. In this particular problem, we are dealing with the tangent function.The tangent function, denoted as \( \tan(x) \), is the ratio of the sine and cosine of an angle. That is, \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). The function is periodic with a period of \( \pi \), meaning it repeats its pattern every \( \pi \) units.In solving equations with tangent, such as \( 3 \tan^2(2x) - 1 = 0 \), we manipulate the equation to isolate the trigonometric term. Here, the equation simplifies to \( \tan^2(2x) = \frac{1}{3} \), highlighting the critical role of understanding trigonometric identities and transformations.

Equation Solving

Solving equations involving trigonometric functions requires a systematic approach. The main goal is to isolate the trigonometric expression and find its possible values. With our given equation, \( 3 \tan^2(2x) - 1 = 0 \), the first step is to simplify. This is achieved by setting \( \tan^2(2x) = \frac{1}{3} \).To find the solutions for \(2x\), we take the square root of both sides. Therefore, \( \tan(2x) = \pm \frac{1}{\sqrt{3}} \). Next, we apply the inverse tangent function to find the angles that satisfy this condition: \( 2x = \arctan(\pm \frac{1}{\sqrt{3}}) \). This results in solutions of \( 2x = \frac{\pi}{6} \) and \( 2x = \frac{5\pi}{6} \).Finally, you divide these results by 2 to solve for \( x \), resulting in the possible solutions \( x = \frac{\pi}{12} \) and \( x = \frac{5\pi}{12} \). Testing these in the original equation confirms they are correct, ensuring the process is thorough and correct.

Angles in Radians

Angles can be measured in degrees or radians. In trigonometry, radians are often preferred because they provide a natural way to describe angles and their relationships with trigonometric functions. A radian is defined as the angle subtended by an arc of a circle that is equal in length to the circle's radius.The conversion between degrees and radians is crucial: \(\pi\) radians equal 180 degrees. This makes it possible to switch between the two by using the formula: \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \). For example, \( \frac{\pi}{12} \) radians equals 15 degrees, and \( \frac{5\pi}{12} \) radians equals 75 degrees.In the context of solving trigonometric equations like \( 3 \tan^2(2x) - 1 = 0 \), angles in radians are often used because formulas involving radians are simpler. As seen, the solutions \( x = \frac{\pi}{12} \) and \( x = \frac{5\pi}{12} \) are naturally expressed in radians, reflecting their efficiency and consistency in trigonometric problem solving.