Problem 12 - Algebra and Trigonometry

Question

\(3-16 \cdot\) Solve the given equation. $$ \tan \theta-3 \cot \theta=0 $$

Step-by-Step Solution

Answer

\( \theta = \frac{\pi}{3} + k\pi \) and \( \theta = \frac{2\pi}{3} + k\pi \), where \( k \) is an integer.

Step 1: Understand the Problem

The equation we need to solve is \( \tan \theta - 3 \cot \theta = 0 \). This trigonometric equation includes \( \tan \theta \) and \( \cot \theta \). Remember that \( \cot \theta = \frac{1}{\tan \theta} \).

Step 2: Express \( \cot \theta \) in terms of \( \tan \theta \)

Using the identity \( \cot \theta = \frac{1}{\tan \theta} \), substitute in the equation: \( \tan \theta - 3(\frac{1}{\tan \theta}) = 0 \).

Step 3: Eliminate the Fraction

To eliminate the fraction, multiply the entire equation by \( \tan \theta \): \( \tan^2 \theta - 3 = 0 \).

Step 4: Solve for \( \tan^2 \theta \)

Rearrange the equation to solve for \( \tan^2 \theta \): \( \tan^2 \theta = 3 \).

Step 5: Find \( \tan \theta \)

Take the square root of both sides: \( \tan \theta = \pm \sqrt{3} \). This gives two possible solutions for \( \theta \).

Step 6: Determine Solutions for \( \theta \)

Solve for \( \theta \) from \( \tan \theta = \sqrt{3} \) and \( \tan \theta = -\sqrt{3} \). This results in \( \theta = \frac{\pi}{3} + k\pi \) and \( \theta = \frac{2\pi}{3} + k\pi \) for integers \( k \).

Step 7: Verify Solutions

Verify by substituting back into the original equation to ensure these expressions satisfy \( \tan \theta - 3 \cot \theta = 0 \). Both solutions satisfy the equation.

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \theta \), is a fundamental concept in trigonometry. It represents the ratio of the sine to the cosine of an angle in a right-angled triangle:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
This function is periodic with a period of \( \pi \), meaning it repeats its values every \( \pi \) radians. The tangent function can take on any real number value, and its graph has distinct features:

• Vertical asymptotes occur at odd multiples of \( \frac{\pi}{2} \), where the cosine function (the denominator) is zero.
• It passes through the origin with the slope scaling upward or downward, depending on the angle.

The tangent function is used to solve trigonometric equations, often requiring simplifying the equation using other trigonometric identities. In this exercise, recognizing how to manipulate \( \tan \theta \) into a square form, \( \tan^2 \theta \), helps solve the equation efficiently. Understanding the behavior and properties of the tangent function aids in determining all possible solutions for an angle \( \theta \).

Cotangent Function

The cotangent function, represented as \( \cot \theta \), is the reciprocal of the tangent function:
\[ \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \]
Like the tangent function, the cotangent is periodic, but with the same period, \( \pi \). This means that every full cycle occurs across \( \pi \) radians too. The cotangent graph features:

• Vertical asymptotes at integer multiples of \( \pi \), where the sine function (the denominator) is zero.
• The graph decreases from positive infinity to negative infinity in each interval between asymptotes.

The cotangent function appears in trigonometric equations, where understanding its reciprocal nature helps reformulate and simplify problems. In the given exercise, expressing \( \cot \theta \) as \( \frac{1}{\tan \theta} \) is key to transforming and solving the equation \( \tan \theta - 3 \cot \theta = 0 \), leading to a quadratic form in terms of \( \tan \theta \).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are always true for any angle. These identities are essential for simplifying complex trigonometric equations and for solving them effectively. Some of the most common identities include:

• Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Reciprocal identities, like \( \tan \theta \cdot \cot \theta = 1 \), which relates tangent and cotangent.
• Quotient identities, stipulating \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

These identities enable us to transform and solve trigonometric equations by applying algebraic manipulations. In our exercise, using the reciprocal identity \( \cot \theta = \frac{1}{\tan \theta} \) directly leads to a reformulation of the equation, allowing efficient reduction and solution. Understanding and being able to use these identities is crucial for solving not just simple but also more complex trigonometric problems, providing a robust toolkit for any calculations involving angles and their functions.