Solving $3 an^3 X = an X$: Step-by-Step Guide

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Hey there, math enthusiasts! Today, we're diving into a trigonometric equation and cracking it step-by-step. The equation we're tackling is $3 an^3 x = an x$, and we need to figure out which of the given options satisfies this equation. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into the solution, let’s make sure we understand what the question is asking. We are given a trigonometric equation, $3 an^3 x = an x$, and a set of possible solutions in degrees: A. $60^{\circ}$, B. $120^{\circ}$, C. $150^{\circ}$, D. $240^{\circ}$, and E. $300^{\circ}$. Our task is to determine which of these angles, when plugged into the equation, makes the equation true. This involves understanding the properties of the tangent function and using algebraic manipulation to simplify the equation.

The tangent function, denoted as $\tan x$, is a fundamental trigonometric function that relates the ratio of the sine and cosine of an angle. Specifically, $\tan x = \frac{\sin x}{\cos x}$. The tangent function has several key properties that are important for solving trigonometric equations. First, it is a periodic function with a period of $180^{\circ}$ or $\pi$ radians, meaning that $\tan(x + 180^{\circ}n) = \tan x$ for any integer $n$. This periodicity is crucial because it means that there are infinitely many solutions to trigonometric equations, but we are only interested in solutions within a certain range in this problem. Second, the tangent function has vertical asymptotes at angles where the cosine function is zero, such as $90^{\circ}$ and $270^{\circ}$, where the function is undefined. Third, the tangent function is positive in the first and third quadrants and negative in the second and fourth quadrants.

When solving trigonometric equations, it’s essential to consider the domain and range of the trigonometric functions involved. For the tangent function, the domain is all real numbers except for angles where $\cos x = 0$, which are $90^{\circ} + 180^{\circ}n$ for integer $n$. The range of the tangent function is all real numbers, meaning it can take any value from $-\infty$ to $\infty$. Understanding these properties helps in identifying extraneous solutions that might arise during the solving process. Additionally, it's crucial to remember the common values of the tangent function for special angles, such as $0^{\circ}$, $30^{\circ}$, $45^{\circ}$, $60^{\circ}$, and $90^{\circ}$, as these frequently appear in trigonometric problems.

Step-by-Step Solution

1. Rearrange the Equation

Let's start by rearranging the equation to make it easier to work with. We have:

$3 an^3 x = an x$

Subtract $\tan x$ from both sides:

$3 an^3 x - an x = 0$

2. Factor out $\tan x$

Now, we can factor out $\tan x$ from the left side of the equation:

$\tan x (3 an^2 x - 1) = 0$

3. Set Each Factor to Zero

For the entire expression to equal zero, at least one of the factors must be zero. So, we set each factor equal to zero:

1. $\tan x = 0$
2. $3 an^2 x - 1 = 0$

4. Solve for $\tan x = 0$

The first equation, $\tan x = 0$, is true when $x$ is an integer multiple of $180^{\circ}$. In other words:

$x = n imes 180^{\circ}$, where $n$ is an integer.

Within the given options, none of the angles directly match this form, but it's crucial to keep this in mind as a potential solution pattern.

5. Solve for $3 an^2 x - 1 = 0$

Now, let's tackle the second equation:

$3 an^2 x - 1 = 0$

Add 1 to both sides:

$3 an^2 x = 1$

Divide by 3:

$\tan^2 x = \frac{1}{3}$

Take the square root of both sides:

$\tan x = \pm \frac{1}{\sqrt{3}}$

6. Find the Angles for $\tan x = \frac{1}{\sqrt{3}}$

The equation $\tan x = \frac{1}{\sqrt{3}}$ corresponds to angles where the tangent is positive. We know that $\tan 30^{\circ} = \frac{1}{\sqrt{3}}$. Since the tangent function has a period of $180^{\circ}$, we can add multiples of $180^{\circ}$ to find other solutions:

• $x = 30^{\circ} + 180^{\circ}n$

For $n = 0$, $x = 30^{\circ}$ (not in the options).

For $n = 1$, $x = 30^{\circ} + 180^{\circ} = 210^{\circ}$ (not in the options).

7. Find the Angles for $\tan x = -\frac{1}{\sqrt{3}}$

The equation $\tan x = -\frac{1}{\sqrt{3}}$ corresponds to angles where the tangent is negative. This occurs in the second and fourth quadrants. The reference angle is still $30^{\circ}$, so we look for angles in the form:

• $x = 180^{\circ} - 30^{\circ} + 180^{\circ}n = 150^{\circ} + 180^{\circ}n$
• $x = 360^{\circ} - 30^{\circ} + 180^{\circ}n = 330^{\circ} + 180^{\circ}n$

Let's check some values:

For $n = 0$, $x = 150^{\circ}$. This is option C.

For $n = 0$ in the second form, $x = 330^{\circ}$ (not in the options).

8. Verify the Solution

To be sure, let's plug $x = 150^{\circ}$ back into the original equation:

$3 an^3(150^{\circ}) = an(150^{\circ})$

We know that $\tan(150^{\circ}) = -\frac{1}{\sqrt{3}}$, so:

$3 \left(-\frac{1}{\sqrt{3}}\right)^3 = -\frac{1}{\sqrt{3}}$

$3 \left(-\frac{1}{3\sqrt{3}}\right) = -\frac{1}{\sqrt{3}}$

$-\frac{1}{\sqrt{3}} = -\frac{1}{\sqrt{3}}$

This confirms that $150^{\circ}$ is indeed a solution.

Analyzing Other Options

While we've found a solution, let’s quickly check why the other options might not work. This can reinforce our understanding of the problem and the tangent function.

• A. $60^{\circ}$: $\tan(60^{\circ}) = \sqrt{3}$. Plugging this into the original equation gives $3(\sqrt{3})^3 = \sqrt{3}$, which simplifies to $9\sqrt{3} = \sqrt{3}$, which is false.
• B. $120^{\circ}$: $\tan(120^{\circ}) = -\sqrt{3}$. Plugging this in gives $3(-\sqrt{3})^3 = -\sqrt{3}$, which simplifies to $-9\sqrt{3} = -\sqrt{3}$, which is also false.
• D. $240^{\circ}$: $\tan(240^{\circ}) = \sqrt{3}$. This is the same tangent value as $60^{\circ}$, so it won't work for the same reasons.
• E. $300^{\circ}$: $\tan(300^{\circ}) = -\sqrt{3}$. This is the same tangent value as $120^{\circ}$, so it also doesn't satisfy the equation.

Conclusion

Therefore, the correct solution to the equation $3 an^3 x = an x$ from the given options is C. $150^{\circ}$. We arrived at this answer by rearranging the equation, factoring, setting each factor to zero, and solving for the possible values of $x$. We also verified our solution by plugging it back into the original equation. Remember, when dealing with trigonometric equations, it's important to consider the periodic nature of trigonometric functions and check for extraneous solutions.

I hope this step-by-step explanation has helped you understand how to solve this type of trigonometric problem. Keep practicing, and you'll become a pro in no time!

For further exploration of trigonometric functions and equations, you might find helpful resources on websites like Khan Academy. Happy solving!