Problem 55 - Precalculus : Building Concepts and Connections

Question

Find the exact values of the given expressions in radian measure. $$\cot ^{-1} \sqrt{3}$$

Step-by-Step Solution

Answer

The exact value in radian measure for \(cot^{-1}(\sqrt{3})\) is \(\pi/6\).

Step 1: Identify Quadrants

Cotangent is positive in the first and third quadrants.

Step 2: Find the reference angle

The \(cot^{-1}\) function will always produce outputs in the range 0 to π. This means that if we are given a positive value, we only need to find the reference angle. Since \(cot(\pi/6) = \sqrt{3}\), therefore \(cot^{-1}(\sqrt{3}) = \pi/6\) by definition.

Step 3: Check the domain

The given value lies in the applicable range of the inverse cotangent function, which is 0 to π, inclusive. Therefore, there are no issues with the domain.

Key Concepts

Cotangent Function

The cotangent function, denoted as \( \cot \), is the reciprocal of the tangent function. In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. Thus, the cotangent is the inverse ratio. Mathematically, \( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}} \). This function has some distinct characteristics. For example, it is undefined when the tangent is zero, which occurs at \( \theta = n\pi \), where \( n \) is an integer.

In the context of inverse trigonometric functions, \( \cot^{-1}(x) \) returns the angle whose cotangent is \( x \). The range for \( \cot^{-1}(x) \) is limited to \( 0 \) to \( \pi \) radians, which are the first and second quadrants on the unit circle. This specific range helps to maintain a one-to-one relationship, ensuring that each cotangent value corresponds to one unique angle.

An important aspect of the cotangent function is identifying where it is positive or negative. It is positive in the first and third quadrants and negative in the second and fourth quadrants. By understanding these sectoral characteristics, solving problems involving the cotangent function becomes much easier.

Reference Angle

A reference angle is the acute angle formed by the terminal side of a standard position angle and the horizontal axis. It is a tool that simplifies the process of finding other related angles. Primarily used in trigonometry, reference angles help in determining the equivalent angles in different quadrants by maintaining the trigonometric function's sign.

To find a reference angle, you take the given angle, whether in degrees or radians, and determine the smallest positive angle to the x-axis. This gives you a practical shortcut when solving trigonometric equations.

For instance, when dealing with \( \cot^{-1}(\sqrt{3}) \), you first identify the angle where this cotangent value occurs. Since \( \cot(\pi/6) = \sqrt{3} \), \( \pi/6 \) is the reference angle that satisfies this inverse function. Reference angles are critical for simplifying complex trigonometry problems and converting angles to manageable forms.

Radian Measure

Radian measure is a way of expressing angles using the radius of a circle. An angle's measure in radians is the length of the arc that it subtends at the center of a circle of units radius. This method of measuring angles is natural and often more convenient than degree measure, particularly in calculus and higher-level mathematics.

In terms of conversion, \( 2\pi \) radians is equivalent to 360 degrees. Therefore, \( \pi \) radians is halfway around a circle and equals 180 degrees. This provides a basis for converting between degrees and radians: \( 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \).

Understanding radian measure is essential for interpreting functions like \( \cot^{-1}(\sqrt{3}) \). This particular function has an angle output in radians, specifically \( \pi/6 \), capturing the essence of radians by representing real angle measures based on the geometry of a circle. Students often switch between these measures, using radians to simplify mathematical calculations while working through trigonometry problems.