Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$\tan x+1=\sqrt{3}+\sqrt{3} \cot x$$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{\pi}{6}, \frac{7\pi}{6}, \frac{2\pi}{3}, \frac{5\pi}{3} \).

Step 1: Transform the Equation

Start with the given equation: \( \tan x + 1 = \sqrt{3} + \sqrt{3} \cot x \). The equation involves both \(\tan x\) and \(\cot x\). Recall that \(\cot x = \frac{1}{\tan x}\). Substitute \( \cot x \) with \( \frac{1}{\tan x} \). We get \( \tan x + 1 = \sqrt{3} + \sqrt{3} \cdot \frac{1}{\tan x} \).

Step 2: Multiply through by \(\tan x\)

To eliminate the fraction, multiply every term by \(\tan x\). This gives us \( \tan^2 x + \tan x = \sqrt{3} \tan x + \sqrt{3} \).

Step 3: Rearrange and Simplify

Rearrange the equation from Step 2 to bring all terms to one side: \( \tan^2 x + \tan x - \sqrt{3} \tan x - \sqrt{3} = 0 \). This simplifies to \( \tan^2 x - (\sqrt{3} - 1) \tan x - \sqrt{3} = 0 \).

Step 4: Solve the Quadratic Equation

This is a quadratic in the variable \( \tan x \), so factorize or use the quadratic formula. For factorizing: assume \( (\tan x - a)(\tan x - b) = 0 \) leads to roots for \( \tan x \). Solving this gives \( \tan x = \frac{\sqrt{3}}{2} \) or \( \tan x = -\frac{\sqrt{3}}{1} \).

Step 5: Solve Trigonometric Equations

For \( \tan x = \frac{1}{\sqrt{3}} \), we find that \( x = \frac{\pi}{6} , \frac{7\pi}{6} \). And for \( \tan x = -\sqrt{3} \), \( x = \frac{2\pi}{3}, \frac{5\pi}{3} \). These solutions are within the interval \([0, 2\pi)\).

Step 6: Verify the Solutions

Verify that values found satisfy the original equation \( \tan x + 1 = \sqrt{3} + \sqrt{3} \cot x \). Plug each value back into the original equation to ensure no mistakes in solving.

Key Concepts

Quadratic Equations

Quadratic equations can often appear in problems that do not initially seem to involve them, like trigonometric equations. A quadratic equation has the general form: \[ ax^2 + bx + c = 0 \] In our exercise, the equation was rearranged into a form that looks like a quadratic: \( \tan^2 x - (\sqrt{3} - 1) \tan x - \sqrt{3} = 0 \). It's essential to recognize this structure so you can apply techniques like factoring or using the quadratic formula. In many cases, especially with trigonometric functions, factoring can simplify the process. When you recognize an equation as quadratic:

• Attempt factoring, considering possible integer or simpler roots.
• If factoring isn't straightforward, use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

This step is crucial in transforming a seemingly complicated problem into a more manageable one. Remember, understanding the structure of the equation helps in finding the solution efficiently.

Trigonometric Identities

Trigonometric identities are powerful tools in solving equations involving trigonometric functions. Recognizing and using these identities helps simplify problems, as seen in converting \( \cot x \) to \( \frac{1}{\tan x} \). It opens the door to potential simplifications. Here are some key identities to remember:

• Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \)
• Reciprocal identities: \( \cot x = \frac{1}{\tan x} \), \( \sec x = \frac{1}{\cos x} \), and \( \csc x = \frac{1}{\sin x} \)
• Co-function identities: \( \sin\left(\frac{\pi}{2} - x\right) = \cos x \) and others.

Such identities can transform complex trigonometric expressions into simpler ones, sometimes leading to quadratics or linear forms. These transformations often make the difference in solving the equation efficiently, guiding you towards solutions by preventing unnecessary complications.

Solution Verification

Solution verification is a critical step that ensures the accuracy of your answers. After solving the equation and finding potential solutions, it's vital to recheck them against the original equation. Here's why it matters:

• Your work might include arithmetic mistakes. Checking ensures that no errors slipped by.
• Certain steps might incorrectly simplify or transform the equation. Verification clarifies if this happened.
• It confirms that solutions fit within the required domain, in this case, the interval \([0, 2\pi)\).

Plugging the solutions back, we find which values make \( \tan x + 1 = \sqrt{3} + \sqrt{3} \cot x \) true. This final proof step solidifies your confidence in the results, securing that they satisfy the initial problem conditions. Remember, consistent practice with verification reinforces the problem-solving skills.