Problem 56 - Precalculus with Calculus Previews

Question

Verify that \(\operatorname{arccot} x=\frac{\pi}{2}-\arctan x,\) for all real numbers \(x\).

Step-by-Step Solution

Answer

The relationship is verified: \(\operatorname{arccot} x = \frac{\pi}{2} - \arctan x\).

Step 1: Understand the Problem

We need to verify if the relationship \(\operatorname{arccot} x=\frac{\pi}{2}-\arctan x\) holds true for all real numbers \(x\). This equation suggests a complementary relationship between the inverse cotangent and inverse tangent functions.

Step 2: Recall Definitions

Recall that \(\operatorname{arccot} x\) is the angle whose cotangent is \(x\) and lies between \(0\) and \(\pi\), while \(\arctan x\) is the angle whose tangent is \(x\) and lies between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).

Step 3: Use Trigonometric Identities

The relationship \(\cot(\theta) = \frac{1}{\tan(\theta)}\) can be utilized here. Therefore, if \(\operatorname{arccot} x = \theta\), then \(\cot(\theta) = x\), meaning \(\theta\) is the angle we are looking for.

Step 4: Substitute and Simplify the Expression

By letting \(\theta = \operatorname{arccot} x\), then \(\tan(\theta) = \frac{1}{x}\). The angle \(\phi = \frac{\pi}{2} - \theta\) would then satisfy the condition \(\tan(\phi) = \cot(\theta) = x\), meaning \(\phi = \arctan x\).

Step 5: Validate the Equation

Substituting back, we have \(\phi = \frac{\pi}{2} - \arctan x\). Therefore, \(\theta = \operatorname{arccot} x = \frac{\pi}{2} - \arctan x\), confirming the original statement.

Key Concepts

Arccotangent

The arccotangent function, denoted as \( \operatorname{arccot} x \), is commonly encountered in various mathematical contexts involving inverse trigonometric functions. It aims to determine the angle \( \theta \) for which the cotangent is equal to \( x \). In simpler terms, if you have an angle whose tangent is the reciprocal of \( x \), then that angle is your arccotangent.

Here are some essential properties and characteristics to remember about the arccotangent:

• Its range is typically between 0 and \( \pi \) radians, which allows it to distinctly identify angles within one complete half of the circle.
• \( \operatorname{arccot} x \) is closely related to the arctangent function by the identity \( \operatorname{arccot} x = \frac{\pi}{2} - \arctan x \).
• This identity reveals a complementary relationship between the arccotangent and the arctangent, both of which effectively describe the connection of cotangent and tangent.

Understanding arccotangent as a function of angles can help bridge connections between various trigonometric concepts, shedding light on the fundamental relationships underpinning inverse trigonometric definitions.

Arctangent

Arctangent, often denoted as \( \arctan x \), is another pivotal inverse trigonometric operation. Specifically, it finds the angle \( \theta \) whose tangent is the given value \( x \). This is incredibly useful when working to deduce angles from known tangent values in triangle and circle problems.

Important features of the arctangent function include:

• The function returns values ranging from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), providing results in the first and fourth quadrants of the unit circle.
• It comes with a reciprocal relationship with cotangent, tying back to the identity \( \arctan x = \frac{\pi}{2} - \operatorname{arccot} x \).

In many mathematical, engineering, and physics problems, arctangent serves a vital role, particularly when calculating angles from slope values or attaining directions from coordinate plane data. Leveraging this function helps simplify complex trigonometric expressions and offers clear insights into angle measurements.

Trigonometric Identities

Trigonometric identities are equations that hold true for all the angles involved. These identities are useful tools in simplifying expressions and solving equations involving trigonometric functions.

In the context of inverse trigonometric functions, certain identities become crucial, such as:

• \( \operatorname{arccot} x = \frac{\pi}{2} - \arctan x \), as used in verifying the complementary relationship between inverse cotangent and tangent.
• Basic identities like \( \tan(\theta) = \frac{1}{\cot(\theta)} \) also support transformations between the functions.

These identities create a rich framework through which trigonometric relationships can be explored and verified. Mastering them allows deeper insight into solving complex problems that involve angles, circles, and periodicity. Notably, these relationships illustrate how inverse functions interact with basic trigonometric principles, ultimately offering a bridge between algebraic and geometric interpretations.