Problem 16 - Precalculus with Limits

Question

In Exercises 11-24, solve the equation. \( 3 \cot^2 x - 1 = 0 \)

Step-by-Step Solution

Answer

The solution to the equation \( 3 \cot^2 x - 1 = 0 \) is given by \( x = \cot^{-1}(± \sqrt{1/3} ) \), where \( x \) is the angle in radians. The exact value can be obtained using a scientific calculator.

Step 1: Rearrange Equation

The first thing that needs to be done is to rearrange the equation to isolate \( \cot^2 x \) on one side of the equation. This is done as follows: \( 3 \cot^2 x - 1 = 0 \) which simplifies to \( 3 \cot^2 x = 1 \), so \( \cot^2 x = 1/3 \).

Step 2: Find \( \cot x \)

Cotangent squared is the square of cotangent, which implies \( \cot x \) can be determined by the square root of \( 1/3 \). So, \( \cot x = ± \sqrt{1/3} \).

Step 3: Apply Inverse Cotangent function

With the value of \( \cot x \), one can now use the inverse cotangent function to find the value of \( x \). This implies that \( x = \cot^{-1}(± \sqrt{1/3} ) \).

Step 4: Final calculation

Finally, use the calculator to compute the specific radian values of \( x \). Make sure that the calculator is in radian mode before undertaking the calculation.

Key Concepts

Cotangent Function

The cotangent function, often denoted as \( \cot x \), is one of the six fundamental trigonometric functions commonly used in mathematics. Understanding the cotangent function is key to solving trigonometric equations, like the one in your exercise.

• Cotangent is the reciprocal of the tangent function. So, \( \cot x = \frac{1}{\tan x} \).
• The function relates to a right triangle as the ratio of the adjacent side to the opposite side, based on the angle \( x \).

Cotangent functions are less intuitive because they are reciprocal, which makes them trickier at first than the sine or cosine. However, once you grasp that they are the inverse of a more commonly understood function, it becomes easier. A common feature of \( \cot x \) is that its graph has vertical asymptotes where \( \tan x = 0 \), specifically at multiples of \( \pi \). This function is periodic with a period of \( \pi \), meaning it repeats its values every \( \pi \) radians.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in trigonometry, allowing us to find angles when given a trigonometric ratio. For our exercise, we focus on the inverse of the cotangent function. This is called the arccotangent function, denoted as \( \cot^{-1} x \).To use the inverse cotangent function, you identify the angle whose cotangent is a given value.

• \( \cot^{-1} x \) finds angle \( x \) such that \( \cot x \) equals the given value.
• Unlike straightforward functions, inverse functions require understanding the range of possible angle values.

While the standard range for \( \cot^{-1} x \) is from \( 0 \) to \( \pi \) (excluding 0 itself since \( \cot 0 \) doesn’t exist), in solving equations, angles might need adjustments to fit specific contexts. These altered ranges depend on whether you are solving within specific intervals or looking for general solutions. Using a calculator or trigonometric tables can help find these angles, but always keep your calculator in radian mode for consistency when working with radians.

Angle in Radians

Radians are a very natural way to measure angles and play a crucial role in trigonometry. Instead of measuring angles in degrees, radians relate directly to the arc length of circles.

• One complete circle is \( 2\pi \) radians.
• Thus, \( \pi \) radians represent 180 degrees, making \( \pi/2 \) radians equivalent to a 90-degree angle.

In trigonometric problems, like your exercise, it's often essential to solve for angles in radians because trigonometric functions are periodic based on radians, not degrees. This means all functions like sine, cosine, and cotangent inherently use radians for their periodic behavior. Also, when your calculator is in radian mode, it facilitates seamless calculations of trigonometric ratios and their inverses consistent with mathematical theory. A good practice is to keep your calculator consistently in radian mode unless specifically working on problems using degrees.