Problem 35 - Algebra and Trigonometry

Question

Find all solutions of the equation. $$\tan ^{5} x-9 \tan x=0$$

Step-by-Step Solution

Answer

Solutions are \( x = n\pi \), \( x = \frac{\pi}{3} + n\pi \), and \( x = \frac{2\pi}{3} + n\pi \) for integers \( n \).

Step 1: Factor the Equation

Firstly, factor the given equation \( \tan^5 x - 9 \tan x = 0 \) by recognizing it as a product of factors:\[\tan x ( \tan^4 x - 9 ) = 0\]This implies that either \( \tan x = 0 \) or \( \tan^4 x - 9 = 0 \).

Step 2: Solve \( \tan x = 0 \)

The equation \( \tan x = 0 \) has solutions where the tangent function is zero, i.e., where the angle \( x \) is an integer multiple of \( \pi \):\[x = n\pi\quad \text{for integers } n\]

Step 3: Solve \( \tan^4 x - 9 = 0 \)

Set the equation \( \tan^4 x - 9 = 0 \) to find where \( \tan^4 x = 9 \). Taking the fourth root of both sides gives:\[\tan^2 x = 3\]Thus, \( \tan x = \pm \sqrt{3} \).

Step 4: Find Solutions for \( \tan x = \sqrt{3} \)

Solve \( \tan x = \sqrt{3} \), which has solutions of:\[x = \frac{\pi}{3} + n\pi\]where \( n \) is an integer.

Step 5: Find Solutions for \( \tan x = -\sqrt{3} \)

Similarly, solve \( \tan x = -\sqrt{3} \), which results in solutions:\[x = \frac{-\pi}{3} + n\pi\]where \( n \) is an integer. Alternatively, it can be expressed as:\[x = \frac{2\pi}{3} + n\pi\]where \( n \) is also an integer.

Key Concepts

Factoring Equations

When dealing with trigonometric equations, factoring is an essential technique for simplifying complex expressions. In the exercise provided, factoring is used to transform the original equation \( an^5 x - 9 an x = 0 \\) into a more manageable form. This is similar to factoring algebraic equations where you look for common factors and simplify the expressions into products of simpler terms. Here:

• Identify common terms: The equation can be rewritten in a way that shows a common factor, which is \( an x\).
• Factor the equation: Pull out the common tangent term, resulting in \( an x ( an^4 x - 9 ) = 0\).
• Separate the factors: The solutions to the equation come from setting each factor equal to zero independently. Thus, you have two separate equations: \( an x = 0\) and \( an^4 x - 9 = 0\).

Factoring is a powerful tool that transforms challenging equations into simpler parts, making them easier to solve. The process relies on the ability to identify and extract common elements within the equation.

Tangent Function

The tangent function, \( \tan x\), is a foundational trigonometric function that relates to the angles and sides of a right-angled triangle. It is periodic with a period of \( \pi\), meaning it repeats its values every \( \pi\) radians. Important characteristics of the tangent function include:

• Zero points: The function equals zero whenever the angle \( x\) is an integer multiple of \( \pi\), that is, \( x = n\pi\), representing critical nodes for solving trigonometric equations.
• Symmetry: The tangent function exhibits odd symmetry, reflected over the origin, such that \( \tan(-x) = -\tan x\).
• Range: Unlike sine and cosine, the range of the tangent function is all real numbers. It can take any value, positive or negative, and is particularly useful in equations that involve radiant slopes.

Understanding how the tangent function behaves is crucial for solving equations involving it, as it helps predict where solutions might occur.

Trigonometric Identities

Trigonometric identities are equalities involving trigonometric functions that hold true for all angle measures. These identities are invaluable for transforming and simplifying expressions. In the exercise, a form of identity is applied implicitly when solving \( \tan^4 x - 9 = 0\), which simplifies to \( \tan^2 x = 3\), breaking down the equation further to \( \tan x = \pm \sqrt{3}\). Important types of identities include:

• Pythagorean identities, such as \( \tan^2 x + 1 = \sec^2 x\), crucial for expressing one trigonometric function in terms of another.
• Angle sum and difference identities, which provide ways to expand \( \tan(a \pm b)\) and verify or simplify complex expressions.
• Even/odd identities, such as telling us \( \tan(-x) = -\tan x\), which help understand the symmetry and periodicity of the function.

By utilizing trigonometric identities effectively, complex trigonometric equations become much more tractable.

Integer Multiples of Pi

In trigonometry, integer multiples of pi (\pi) play a pivotal role, especially in finding solutions for equations involving the tangent function. The angles at which trigonometric functions return to zero or their starting values are often these multiples. For example:

• Solving \( \tan x = 0\) relies on finding when the tangent function resets, which happens at \( x = n\pi\) (where \( n\) is any integer). These are solutions because tangent is zero at these points.
• Similarly, if the angle results in values like \( \tan x = \sqrt{3}\) or \( \tan x = -\sqrt{3}\), further integer multiples of \( \pi\) are added to ensure all possible solutions are captured. E.g., \( x = \pi/3 + n\pi\) for \( \tan x = \sqrt{3}\).
• The reliance on \( \pi\) in solutions showcases the function’s periodicity, pivotal in understanding how trigonometric functions loop back in predictable cycles.

Recognizing the importance of integer multiples of \( \pi\) allows for more holistic solution sets to trigonometric equations, tapping into the inherent periodic cycles of these functions.