Problem 42 - Algebra 2

Question

Solve each equation for \(0 \leq \theta<2 \pi\) $$ \cot \theta=-10 $$

Step-by-Step Solution

Answer

The solutions for the equation are \( \theta_1 = \pi - \cot^{-1} (10) \) and \( \theta_2 = 2\pi - \cot^{-1} (10) \), where \(\cot^{-1} (10) \) represents the acute angle cotangent is equal to 10.

Step 1: Write the cotangent function as a ratio

Let's express the cotangent function as the ratio of cosine to sine functions. Therefore, \( \cot \theta = \cos \theta / \sin \theta = -10 \)

Step 2: Solve for cotangent in each quadrant

Since the cotangent function is negative in the second (from \(\pi/2\) to \(\pi\)) and the fourth (from \(3\pi/2\) to \(2\pi\)) quadrants, these will be the valid values of \(\theta\). Remember, from the unit circle value, cotangent is negative when sine and cosine have opposite signs.

Step 3: Find the main solution

We first deal with the unit circle where the angle is acute, namely the first quadrant. At this point, set \(\cot \theta = 10\), and solve for \(\theta\). The solution can be found using an inverse cotangent function, \(\theta = \cot^{-1} (10) \)

Step 4: Get solutions for \(\theta\) in other quadrants

Transposing the obtained solution from the first quadrant to second and fourth quadrants (where cotangent is negative). The \(\theta\) value for the second quadrant is \(\pi - \cot^{-1} (10)\) and the fourth quadrant is \(2\pi - \cot^{-1} (10)\)

Step 5: Verify the solutions

Check if the found solutions for \(\theta\) are within the valid interval \(0 \leq \theta < 2 \pi\). If they are, then these are the solutions for the equation.

Key Concepts

Cotangent Functions

The cotangent function, denoted as \( \cot \theta \), is one of the six primary trigonometric functions. Unlike sine or cosine, it is defined as the ratio of the cosine and sine of an angle. Specifically, \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). This function is the complement of the tangent function, which is the ratio of sine to cosine.When solving problems involving the cotangent function, it is important to consider its properties:

• It is undefined when the sine of the angle is zero, as this results in division by zero.
• The cotangent function has a period of \( \pi \), meaning it repeats its values every \( \pi \) radians.
• Its values range from \(-\infty\) to \(+\infty\).

In the context of the given exercise, you need to understand that cotangent being negative means one of the sine or cosine values must be negative. This characteristic helps determine the solutions within the appropriate quadrants.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when given specific trigonometric values. The inverse of the cotangent function is denoted as \( \cot^{-1} (x) \). This function will return an angle whose cotangent is \( x \).

• It is important to note that \( \cot^{-1} (x) \) for positive values will yield angles in the first quadrant (\(0\) to \(\frac{\pi}{2}\)).
• When dealing with negative values, the values often fall in the second and fourth quadrants when transposing the primary angle.

For instance, if you find \( \theta \) using \( \cot^{-1} (10) \), you need to remember the placement and characteristics of the angle in various quadrants to ensure a complete solution. This allows for the conversion of angles when considering their innate negatives.

Unit Circle

The unit circle is a crucial concept for understanding trigonometric functions. It is a circle with a radius of one unit centered at the origin of a coordinate plane. This circle provides a visual representation of angle measures and their corresponding trigonometric values.

• Angles are measured in radians, moving counter-clockwise from the positive x-axis.
• Each point on the unit circle represents the cosine and sine values of an angle \( \theta \), formatted as \((\cos \theta, \sin \theta)\).
• The unit circle allows for quick interpretation of trigonometric functions due to the predictable pattern of sine, cosine, and tangent values.

In the exercise, identifying where the cotangent function is negative involves analyzing the unit circle to determine when sine and cosine have opposite signs. This insight directly ties into quadrant analysis, enhancing one's ability to solve trigonometric equations efficiently.

Quadrant Analysis

Quadrant analysis is essential when solving trigonometric functions as it helps determine where a function is positive or negative. Understanding each quadrant's particular characteristics can simplify the resolution of equations involving trigonometric ratios.

• The first quadrant \((0, \frac{\pi}{2})\): All trigonometric functions are positive.
• The second quadrant \((\frac{\pi}{2}, \pi)\): Sine is positive, while cosine and tangent are negative, making cotangent also negative.
• The third quadrant \((\pi, \frac{3\pi}{2})\): Both sine and cosine are negative, but tangent and cotangent are positive.
• The fourth quadrant \((\frac{3\pi}{2}, 2\pi)\): Cosine is positive, sine is negative, thus making cotangent negative.

In this exercise, knowing that the cotangent is negative points you to the second and fourth quadrants. By breaking down each quadrant's sign rules, you can confidently find all potential solutions for \(\theta\) within the exercise's constraints (\(0 \leq \theta < 2\pi\)).