Problem 14 - Calculus Single Variable

Question

Calculate the value of the given inverse trigonometric function at the given point. $$ \operatorname{arccot}(-\sqrt{3}) $$

Step-by-Step Solution

Answer

\( \operatorname{arccot}(-\sqrt{3}) = 5\pi/6 \).

Step 1: Understand the Definition of Arccot

The function \( \operatorname{arccot}(x) \) returns the angle \( \theta \) whose cotangent is \( x \). The principal value of \( \operatorname{arccot} \) is typically defined in the range \( (0, \pi) \).

Step 2: Identify the Cotangent Value

Given \( \operatorname{arccot}(-\sqrt{3}) \), we need to find an angle \( \theta \) such that \( \cot(\theta) = -\sqrt{3} \).

Step 3: Recall Cotangent Function Values

Cotangent has the reciprocal values of tangent, so we remember that \( \cot(\pi/6) = \sqrt{3} \). Thus, we need an angle where \( \cot(\theta) = -\sqrt{3} \).

Step 4: Determine the Quadrant

Since we need \( \cot(\theta) = -\sqrt{3} \) and the range of \( \operatorname{arccot} \) is \((0, \pi)\), the angle \( \theta \) must be in the second quadrant where cotangent is negative.

Step 5: Find the Specific Angle

In the second quadrant, the reference angle corresponding to \( \pi/6 \) is \( \pi - \pi/6 = 5\pi/6 \). This angle satisfies \( \cot(\theta) = -\sqrt{3} \).

Step 6: Conclude the Solution

Thus, the value of \( \operatorname{arccot}(-\sqrt{3}) \) is \( 5\pi/6 \).

Key Concepts

Arccotangent

Arccotangent, often written as \( \operatorname{arccot}(x) \), is an inverse trigonometric function. It gives us the angle whose cotangent equals a given number. Understanding arccotangent requires knowing the cotangent function, which is the inverse of the tangent function. In simple terms, this means that \( \operatorname{arccot}(x) \) returns the angle \( \theta \) such that \( \cot(\theta) = x \).
Here's a breakdown of key points:

• The cotangent of an angle is the ratio of the adjacent side to the opposite side in a right-angled triangle.
• Therefore, if \( \operatorname{arccot}(x) \) gives us \( \theta \), then \( \theta \) is the angle where this ratio is exactly \( x \).
• Like other inverse trigonometric functions, arccotangent helps in transitioning from a ratio back to an angle size.

When using arccotangent, it's essential to know which angles correspond to specific values of cotangent within the function's restricted range, as inverses are only defined in certain intervals to ensure they produce one value.

Principal Value

The concept of principal value is fundamental when dealing with inverse trigonometric functions like arccotangent. The principal value is essentially the main range in which the function returns outputs, ensuring that for each value input, there's a unique angle output in this restricted domain.
For \( \operatorname{arccot}(x) \), the principal value is defined within the range \((0, \pi)\). This means that for any input value \( x \), the function will provide an angle \( \theta \) where:

• \( 0 < \theta < \pi \).
• The range excludes the endpoints, stopping the function from producing angles of \( 0 \) or \( \pi \).

This ensures a unique angle is determined for each input by restricting the domain. This is similar to ensuring the answers to a question on an exam are within a specific range - straightforward and predictable.

Trigonometric Identities

Understanding trigonometric identities is crucial for solving problems with inverse trigonometric functions.Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables where they are defined. They are a toolkit for simplifying expressions and solving equations.
Here are some critical aspects:

• The identity \( \cot(\theta) = \frac{1}{\tan(\theta)} \) shows the reciprocal relationship with the tangent function.
• The identity \( \tan(\pi/6) = \frac{1}{\sqrt{3}} \) helps derive \( \cot(\pi/6) = \sqrt{3} \).

These identities are essential when working with arccotangent because they help identify the respective angle, by relating different trigonometric functions together. They serve as a bridge, helping us to connect known values within various trigonometric relationships.

Quadrants

When working with trigonometric functions, understanding quadrants helps determine the angle's sign and direction. The coordinate plane is divided into four quadrants, each displaying different sign properties for sine, cosine, and tangent functions.
The quadrant details are:

• First Quadrant: All trigonometric ratios are positive.
• Second Quadrant: Sine is positive, but cosine and tangent are negative.
• Third Quadrant: Tangent is positive, while sine and cosine are negative.
• Fourth Quadrant: Cosine is positive, but sine and tangent are negative.

In the context of \( \operatorname{arccot}(-\sqrt{3}) \), knowing that the function's value must fall in the range \( (0, \pi) \) indicates it will be in the second quadrant because the cotangent is negative there. Recognizing these quadrant characteristics forms a basis for determining the correct angle and helps ensure the answer fits within the principal range.